Algebra and Logic, Vol. 42, No.

3, 2003

SUPERLOCALS IN SYMMETRIC
AND ALTERNATING GROUPS
D. O. Revin∗ UDC 512.542

Key words: symmetric group, alternating group, p-superlocal.

On Aschbacher’s definition, a subgroup N of a finite group G is called a p-superlocal for a prime

p if N = NG (Op (N )). We describe the p-superlocals in symmetric and alternating groups,
thereby resolving part way Problem 11.3 in the Kourovka Notebook [3].

A basis for studying p-local subgroups of every Lie-type group G over a finite field of characteristic p is
the known Borel–Tits theorem (cf. [1, Sec. 3.12]). This says that, for any subgroup H of G, there exists
a parabolic subgroup R such that H ≤ R and Op (H) ≤ Op (R); hereinafter, Op (G) denotes the greatest
normal p-subgroup of G. A characterization of parabolic subgroups in terms of subgroups that coincide
with normalizers of their greatest normal p-subgroups is readily obtained as a consequence of the theorem
cited. The concept of a p-superlocal in an arbitrary finite group introduced in [2] is a generalization of the
property of parabolic subgroups mentioned.
Definition. Let G be a finite group and p a prime. A subgroup N of G is called a p-superlocal in G if
N = NG (Op (N )).
Aschbacher noticed that every p-local subgroup H of a finite group G (i.e., the normalizer in G of some
p-subgroup) is contained in the p-superlocal N such that Op (H) ≤ Op (G). This conception of superlocals
admits further development.
Definition. Let G a finite group, p a prime, and H1 and H2 subgroups of G. We say that H1 is
p-contained in H2 (written H1 ≤p H2 ) if H1 ≤ H2 and Op (H1 ) ≤ Op (H2 ). It is easy to see that the relation
≤p induces a partial order on the set of subgroups. Maximal elements under this order are called p-maximal
subgroups of G.
Clearly, every p-maximal subgroup is also a p-superlocal. Below, in Proposition 3, we show that p-
superlocals and p-maximal subgroups are in fact same objects. Therefore a complete analog of the Borel–
Tits theorem fits well with the former.
It would be natural to introduce a concept of a p-radical, dual to the concept of a p-superlocal.
Definition. Let G be a finite group and p a prime. We call a subgroup P a p-radical in G, or a
radical p-subgroup (by analogy with unipotent radicals of parabolic subgroups in Lie-type groups), if P is
a p-subgroup of G, and P = Op (NG (P )).
∗ Supported
by RFBR grant No. 02-01-00495; by FP “Universities of Russia” grant UR.04.01.031; by the 6th Contest–
Expertise of the RAS Young Scientists Projects in Fundamental and Applied Research (1999), grant No. 7; by a grant of
Lavrentiev Contest for Young Scientists under SO RAN; and by the Council for Grants (under RF President) and State Aid
of Fundamental Science Schools, grant NSh-2069.2003.1.

Translated from Algebra i Logika, Vol. 42, No. 3, pp. 338-365, May-June, 2003. Original article submitted August 30,
2001.

192 0002-5232/03/4203-0192 $25.00

c 2003 Plenum Publishing Corporation
 Clearly, there is a one-to-one correspondence between p-superlocals and p-radicals of G. Namely, if
N is a p-superlocal then Op (N ) is its corresponding p-radical. And if P is a p-radical then NG (P ) is its
corresponding p-superlocal. In particular, we may speak of a radical of a p-superlocal. In essence, describing
p-superlocals and describing p-radicals in some finite group are an equivalent problem.
Still in [2] are the issues that Aschbacher thinks of as most essential at a “postclassification” stage.
Among these is the problem of obtaining a description of p-superlocals in alternating groups, and in Lie-
type groups. The problem is closely connected with issues of the local analysis playing a key part in proving
the classification theorem. In 1991, Aleev re-worded this in [3, Problem 11.3].
Note that radical subgroups of some sporadic groups have been described in [4-9].
The objective of the present article is to furnish a description of p-superlocals, and of radical p-subgroups,
in alternating groups, and thereby resolve in part the above-mentioned problem. Since every p-superlocal
in an alternating group An is the intersection of An and some superlocal in a symmetric group Sn (cf.
Thm. 3 below), our main concern is to describe the p-superlocals in symmetric groups. Under Sec. 1, we
list a number of elementary useful properties of p-superlocals.
Before formulating our main results, we make some terminological and notational conventions. The
bulk of the notation is basically standard. All groups and sets are assumed finite. For every set M , |M |
denotes the cardinality of M . Throughout, p stands for some prime, and the words a “superlocal” and a
“radical” denote a p-superlocal and a p-radical, respectively. If V is a vector space over some field then V #
denotes the set {v ∈ V | v = 0}.
We always write Ω for some set, and denote by S(Ω) and A(Ω) a symmetric group and an alternating
group on Ω. Elements of the set Ω are called points. If G acts on Ω, and x ∈ Ω, then Gx denotes a group
{g ∈ G | xg = x}, called the stabilizer of a point x in G. We say that G acts regularly on Ω if G acts on
Ω transitively, and the stabilizer in G of every point from Ω is trivial. Such an action is equivalent to the
action of G by right shifts on the set of its elements. If Ω1 ⊆ Ω then we identify S(Ω1 ) with the point
stabilizer

S(Ω)Ω\Ω1 = S(Ω)x
x∈Ω\Ω1

of the complement Ω \ Ω1 to the set Ω1 .

We use Epα to denote an elementary Abelian p-group of order pα . By the Cayley theorem, the group
Epα is embedded as a transitive subgroup in S(Ω) if |Ω| = pα . We identify Epα with this subgroup of S(Ω).
Let Ω1 , . . . , Ωs ⊆ Ω, Ωi ∩ Ωj = ∅ for i = j, and Hi ≤ S(Ωi ), i = 1, . . . , s. Then [Hi , Hj ] = 1 for i = j. A
product of the subgroups H1 , . . . , Hs is said to be independent. Clearly, the independent product is direct.
On [2], a full diagonal subgroup in the direct product G = G1 × · · · × Gm of the groups G1 , . . . , Gm is the
subgroup H ≤ G such that a restriction of the coordinate projection πi : G → Gi to H is an isomorphism
of the groups H and Gi , i = 1, . . . , m.
Definition. Letting Ω1 ⊆ Ω and H ≤ S(Ω1 ), we assume that there exist elements g1 = 1, g2 , . . . , gm ∈
S(Ω) such that Ω1 gi ∩ Ω1 gj = ∅ for i = j. Then H gi ≤ S(Ω1 gi ) for any i = 1, . . . , m. A subgroup
H g1 H g2 . . . H gm of S(Ω) is called an mth independent degree of the group H.
Definition. Let G = S(Ω), where |Ω| = pα , and (α1 , . . . , αk ) be an ordered tuple of natural numbers so
that α1 +· · ·+αk = α. If k = 1 then a transitive elementary wreath product of type (α) in G is any transitive
elementary Abelian subgroup. Let k > 1. In order to define a transitive elementary wreath product of
type (α1 , . . . , αk ), we take some partition Ω = Ω1 ∪ · · · ∪ Ωpαk of the set Ω into a union of disjoint subsets
Ωi = {xi1 , . . . , xipα−αk }, i = 1, . . . , pαk , of equal power whose elements will be numbered, in a way. Let P1

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be a transitive elementary wreath product of type (α1 , . . . , αk−1 ) in the group S(Ω1 ). The subgroup

S = (x11 xi1 )(x12 xi2 ) . . . (x1pα−αk xipα−αk ) | i = 2, . . . , pαk 

of G is isomorphic to a group Spαk and acts transitively and faithfully on the set ∆ = {Ω1 , . . . , Ωpαk }; so,
we can identify it with S(∆). (Such a choice of the subgroup in G is of course not uniquely determined: it
depends on how elements of each of the sets Ωi are numbered.) Let E be a transitive elementary Abelian
subgroup of S. Put  g
P =E P1 .
g∈E

The subgroup P is called a transitive elementary wreath product of type (α1 , . . . , αk ). In this case P
Epα1  Epα2  · · ·  Epαk .
Remark. In the notation of the previous definition,
 g
P0 = P1  P, P0 = P1 × · · · × Ppαk ,
g∈E

where Pi ≤ S(Ωi ) and Pi = P1gi for some element gi ∈ E, i = 1, . . . , pαk . Note also that the subgroup P0 is
normalized by the subgroup S. In fact, by Lemma 4 (cf. Sec. 2), S = SΩi E holds for any i = 1, . . . , pαk ,
where SΩi is the stabilizer of Ωi ∈ ∆. The group SΩi centralizes S(Ωi ); consequently, it also centralizes
Pi and the group E acts on the set {P1 , . . . , Ppαk } by conjugations. Hence S too acts on {P1 , . . . , Ppαk },
and so it normalizes P0 . Moreover, every subgroup of G conjugate to P is a transitive elementary wreath
product of the same type as is P .
The main result of the present article is a combination of the following three theorems.
THEOREM 1. Let G = S(Ω). A non-trivial p-subgroup P of G is a radical of some transitive
superlocal N of G if and only if, for some natural m, P is an mth independent degree of some transitive
elementary wreath product P1 of type (α1 , . . . , αk ) in some group S(Ω1 ), Ω1 ⊆ Ω, and the following
conditions hold:
(1) m|Ω1 | = mpα1 +···+αk = |Ω|;
(2) if p = 2 and α1 = · · · = αk = 1 then m = 2, 4.
In this event N NS(Ω1 ) (P1 )  Sm and NS(Ω1 ) (P1 ) is a split extension of the group P1 Epα1  Epα2 
· · ·  Epαk by the group NS(Ω1 ) (P1 )/P1 GLα1 (p) × GLα2 (p) × · · · × GLαk (p).
THEOREM 2. Let G = S(Ω) and N be a subgroup of G. N is a superlocal in G if and only if
there exists a partition of the set Ω into a union of mutually disjoint subsets Ω1 , . . . , Ωs such that N is
a product of the subgroups N1 , . . . , Nm , each of which is a transitive superlocal in a corresponding group
S(Ω1 ), . . . , S(Ωm ), and the following conditions hold:
(1) among Pi = Op (Ni ), i = 1, . . . , m, there is at most one trivial group;
(2) if i = j and Pi and Pj are non-trivial subgroups then the transitive elementary wreath products
whose independent degrees are Pi and Pj (cf. Thm. 1) are of different types.
THEOREM 3. Let S = S(Ω) and A = A(Ω). Then:
(1) Assume NA ≤ A. The subgroup NA is a p-superlocal in A iff NA = A ∩ NS , where NS is some
p-superlocal in S satisfying one of the following conditions:
(1.1) Op (NS ) ≤ A;
(1.2) the subgroup A ∩ Op (NS ) is not a p-radical in S.

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 (2) Suppose PA is a p-subgroup of A. Then PA is a p-radical in A iff PA = A ∩ PS , where PS is some
p-radical in S.

1. GENERAL PROPERTIES OF SUPERLOCALS

A basis for studying the properties of superlocals is the following:

Proposition 1. If a superlocal N of a group G normalizes a p-subgroup Q then Q ≤ Op (N ). In
particular, Op (G) ≤ Op (N ).
Proof. Let P = Op (N ). Then N normalizes a subgroup P Q, and hence NP Q (P ) = P Q ∩ N  N .
Consequently, P ≤ NP Q (P ) ≤ Op (N ) = P , whence NP Q (P ) = P . If Q ≤ P , then P < P Q, and by a
known property of nilpotent groups, we have P < NP Q (P ), which is a contradiction. The proposition is
proved.
Obviously, the following holds:
Proposition 2. If N is a superlocal in G, and N ≤ H ≤ G, then N is a superlocal in H.
That the concepts of a superlocal and of a p-maximal subgroup are equivalent is shown in the next
proposition.
Proposition 3. Let N1 and N2 be superlocals in G and P1 = Op (N1 ) and P2 = Op (N2 ) be the
corresponding radicals. Suppose also that N1 ≤ N2 . Then P1 ≥ P2 and P2  N1 . If, in addition, N1 ≤p N2 ,
then N1 = N2 . In particular, every superlocal is a p-maximal subgroup.
Proof. Propositions 1 and 2 imply that P1 ≥ P2 , and since N1 is contained in the normalizer of the
group P2 , the subgroup P2 is normal in N1 . If N1 ≤p N2 then P1 = Op (N1 ) ≤ OP (N2 ) = P2 . In this way
P1 = P2 , and so therefore N1 = NG (P1 ) = NG (P2 ) = N2 . Since any p-maximal subgroup is a superlocal,
every superlocal coincides with the p-maximal subgroup in which it is p-contained. The proposition is
proved.
Proposition 4. Let H be a normal subgroup of G. Then the following statements hold:
(1) If P is a radical in G then P ∩ H is one in H.
(2) If PH is a radical in H and NH = NH (PH ) is the corresponding superlocal, then N = NG (PH ) is a
superlocal in G such that N ∩ H = NH , and P = Op (N ) is a radical in G for which P ∩ H = PH .
Proof. (1) Let P be a radical in G and N = NG (P ) be the corresponding superlocal. It is easy to
see that N normalizes each of the subgroups P ∩ H, NH (P ∩ H), and Op (NH (P ∩ H)). By Proposition 1,
Op (NH (P ∩ H)) ≤ P ∩ H, and so P ∩ H is a radical in H.
(2) Let PH be a radical in H and NH = NH (PH ) be the corresponding superlocal. Suppose N = NG (PH )
and P = Op (N ). Since NH ≤ N , the group NH normalizes groups P and P ∩ H. In view of Proposition 1,
P ∩ H ≤ PH . The inverse inclusion is obvious, and so P ∩ H = PH . It is also easy to see that NG (P )
normalizes a subgroup P ∩ H = PH , and consequently NG (P ) ≤ NG (PH ) = N . Hence NG (P ) = N , and
so N is a superlocal, and P is a radical, in G. In this case N ∩ H = NH (PH ) = NH . The proposition is
proved.
Proposition 5. Let G = G1 × · · · × Gn . The subgroup P (or N ) is a radical in G (resp., a superlocal
in G) if and only if each of the subgroups P ∩ Gi (or N ∩ Gi ) is a radical (resp., a superlocal) in Gi ,
i = 1, . . . , n. Moreover, P = (P ∩ G1 ) × · · · × (P ∩ Gn ) [N = (N ∩ G1 ) × · · · × (N ∩ Gn )].
Proof. Let πi : G → Gi be a map by which every element of G is assigned its coordinate projection
onto Gi , i = 1, . . . , n. Let P be a radical in G and N = NG (P ) be the corresponding superlocal. Since
[Gi , Gj ] = 1 for 1  i, j  n, and i = j, P πi is a normal subgroup in the product N π1 . . . N πn . On the

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other hand, the group N is contained in N π1 . . . N πn , and so it normalizes each of the subgroups P πi ,
i = 1, . . . , n. In virtue of Proposition 1, P π1 . . . P πn ≤ P . The inverse inclusion also holds of course. Hence
P = P π1 . . . P πn and P πi ≤ P ∩ Gi , i = 1, . . . , n. Since πi acts identically on Gi , i = 1, . . . , n, we have
P ∩ Gi = (P ∩ Gi )πi ≤ P πi for any i. This means that P = (P ∩ G1 ) . . . (P ∩ Gn ). In view of Proposition 4,
the subgroup P ∩ Gi is a radical in Gi , i = 1, . . . , n. It is also clear that

N = NG (P ) = NG1 (P ∩ G1 ) . . . NGn (P ∩ Gn ),

and N ∩ Gi = NGi (P ) = NGi (P ∩ Gi ) is a superlocal in Gi for any i = 1, . . . , n.

Conversely, if P1 , . . . , Pn are radicals in G1 , . . . , Gn and N1 = NG1 (P1 ), . . . , Nn = NGn (Pn ) are the
corresponding superlocals, then
NG (P1 . . . Pn ) = N1 . . . Nn ,
Op (NG (P1 . . . Pn )) = Op (N1 ) . . . Op (Nn ) = P1 . . . Pn .
Therefore P = P1 . . . Pn is a radical in G and N = N1 . . . Nn is a superlocal in G. Moreover, Pi = P ∩ Gi
and Ni = N ∩ Gi , i = 1, . . . , n. The proposition is proved.
In some cases we may assert that the images of a radical and of a superlocal in a factor group will be a
radical and a superlocal, respectively. Clearly, the following holds:
Proposition 6. Let H be a normal subgroup of G, so that either H is a p-subgroup, or H ≤ Z(G).
The subgroup N (or P ) is a superlocal (resp., a radical) in the group if and only if H ≤ N (Op (H) ≤ P )
and the subgroup N/H (or P H/H) is a superlocal (resp., a radical) in G/H.

2. PRELIMINARY LEMMAS

LEMMA 1. Assume that G contains a normal subgroup A and a subgroup B for which A ∩ B = 1
and AB = G, letting H ≤ B. Then CG (H) = CA (H)CB (H) and NG (H) = CA (H)NB (H).
Proof. Clearly, CA (H)CB (H) ≤ CG (H) and CA (H)NB (H) ≤ NG (H). Moreover, CA (H) = NA (H)
since H normalizes NA (H), and NA (H) ∩ H ≤ A ∩ B = 1. We have

NG (H)/NA (H) = NG (H)/(A ∩ NG (H)) ANG (H)/A ≤ NG/A (AH/A) NB (H),

whence
|NG (H)| ≤ |NA (H)||NB (H)| = |NA (H)NB (H)| and NG (H) = NA (H)NB (H).
Similarly, CG (H) = CA (H)CB (H). The lemma is proved.
Below are two obvious results.
LEMMA 2. Let G = AB, where A and B are subgroups of G such that A  G, A ∩ B = 1, and
Op (B) = 1. Then Op (G) = Op (A).
LEMMA 3. A subgroup Op (Sm ) is distinct from the trivial if and only if one of the following holds:
(1) p = 2 and m = 2;
(2) p = 3 and m = 3;
(3) p = 2 and m = 4.
In these cases Op (Sm ) is a transitive elementary Abelian subgroup of Sm .
LEMMA 4. Let π : G → S(Ω) be a permutation representation of degree n and A be an Abelian
subgroup of G acting faithfully and transitively on Ω. Assume also that A ≤ H ≤ G. Then one of the
statements below holds:

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 (1) Ax = 1 for any point x ∈ Ω;
(2) |A| = n;
(3) H = AHx ;
(4) if H acts on Ω faithfully then CH (A) = A;
(5) if G = S(Ω) and π is a natural representation then every subgroup of G that is isomorphic to A and
acts transitively on Ω is conjugate to A.
Proof. (1) Let x ∈ Ω. Since A acts on Ω transitively, the subgroups Ax and Ay are conjugate in A
for any point y ∈ Ω, and A being an Abelian group implies that Ax = Ay ; consequently, Ax acts on Ω
identically. By assumption, A acts on Ω faithfully, and so Ax ≤ A ∩ ker π = 1.
(2) Since the subgroup A acts on Ω transitively, n = |Ω| = |A : Ax | = |A|.
(3) We also have Hx ∩ A = Ax = 1, and so |Hx A| = |Hx ||A|. On the other hand, since A ≤ H, the
group H acts transitively on Ω, and |A| = n = |H : Hx | = |H|/|Hx |; therefore, |H| = |Hx ||A| = |Hx A|.
Hence H = Hx A.
(4) Suppose that H acts faithfully on Ω and that C = CH (A). Then C = Cx A by (3). Since C acts on
Ω transitively, for any y ∈ Ω, there exists an element g ∈ C such that Cy = Cxg . Let g = ca, where c ∈ Cx
and a ∈ A. Then Cy = Cxca = Cxa = Cx since a centralizes Cx . This means that Cx acts trivially on Ω, and
in view of the action of H being faithful, the subgroup Cx is trivial. Thus C = A.
(5) Now let G = S(Ω), π be a natural representation, and G ≥ A1 A, where A1 is a transitive
permutation group. Since the stabilizer of a point in A, and in A1 , is trivial, the corresponding permutation
representations for A and for A1 are equivalent to their regular representations. Therefore the isomorphism
ϕ : A → A1 , too, can be treated as some representation of A equivalent to the natural. Thus there exists
a one-to-one map g : Ω → Ω such that aϕ = ag for any a ∈ A. Since g ∈ G, we have A1 = Aϕ = Ag . The
lemma is proved.
LEMMA 5. Let E be an elementary Abelian group of order pα acting transitively and faithfully on
Ω. Suppose also that E, treated as a vector space over a field GF(p), is decomposable into a direct sum of
subspaces U and V . Then:
(1) the group V acts transitively and regularly on a set of orbits of the group U ;
(2) an intersection of any orbit of the group U and any orbit of the group V contains exactly one point,
and every point of Ω belongs to the intersection of some orbits of U and V .
Proof. (1) Since U  E, every subgroup of E acts on the set of orbits of U ; moreover, the group E
acts on Ω transitively, and so E also acts transitively on the set of orbits of U . Let Ω1 and Ω2 be such two
orbits. There exists an element g ∈ E for which Ω1 g = Ω2 . Let g = uv, where u ∈ U and v ∈ V . Then

Ω2 = Ω1 g = Ω1 uv = Ω1 v,

that is, the group V acts transitively on the set of orbits of U . Let x ∈ Ω1 . Since

Ux ≤ Ex = 1 and |Ω1 | = |U : Ux | = |U |,

the number of orbits of U is equal to

|Ω|/|Ω1 | = |E|/|U | = |V |.

Since V acts transitively on the set of orbits of U , the stabilizer in V of every such orbit has index |V |,
and its order is, therefore, equal to 1. This means that V acts regularly on the set of orbits of U .

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 (2) Let ΓU and ΓV be sets of orbits of the groups U and V , respectively. Assume ΩU ∈ ΓU , ΩV ∈ ΓV ,
and x ∈ ΩV . Since V acts transitively on ΓU , xv ∈ ΩU for some v ∈ V . We have xv ∈ ΩV , and so the
intersection ΩU ∩ ΩV is non-empty. Obviously, every point of Ω lies in some intersection of the orbits of U
and V ; therefore,
|Ω|  |ΩU ∩ ΩV |  |Ω|.
ΩU ∈ΓU , ΩV ∈ΓV

Hence every such intersection contains exactly one point. The lemma is proved.
LEMMA 6. Let G act on Ω and |Ω|
2. Then its Frattini subgroup Φ(G) cannot act transitively
on Ω.
Proof. Assume that Φ(G) acts transitively on Ω, and hence G too acts transitively. Let x ∈ Ω. Then

|G : Gx | = |Ω| = |Gx Φ(G) : Gx |.

Hence G = Gx Φ(G), and by a known property of Frattini subgroups, G = Gx . We have arrived at a

contradiction with the transitivity of G. The lemma is proved.
LEMMA 7. Suppose that G contains a normal subgroup X = X1 × · · · × Xs such that all subgroups
Xi , i = 1, . . . , s, are conjugate in G. Assume also that G = HX, where H is an Abelian subgroup for which
H ∩ X = 1. Then the following statements hold:
(1) [H, X] ≤ G ≤ X and the coordinate projection of the subgroups [H, X] and G onto a subgroup Xi
coincides with Xi , i = 1, . . . , s;
(2) if G is a p-group and H is an elementary Abelian group then Φ(G) ≤ X and the coordinate projection
of the subgroup Φ(G) onto Xi coincides with Xi , i = 1, . . . , s.
Proof. If X = 1, then both of the statements above are obvious, and we so assume that X = 1.
(1) Let πi : X → Xi be a map of the coordinate projection, i = 1, . . . , s. Since G/X H is an Abelian
group, [H, X] ≤ G ≤ X. Let i ∈ {1, . . . , s} and xi ∈ Xi . There exists an element h ∈ H such that
Xih = Xj = Xi . Consequently,
[h, xi ]πi = ((x−1 h
i ) xi )πi = xi .

Therefore [H, X]πi = G πi = Xi .

(2) If G is a p-group and G/X H is elementary Abelian then Φ(G) ≤ X. Since G ≤ Φ(G), for any
i = 1, . . . , s we have
Xi = G πi ≤ Φ(G)πi ≤ Xi .
The lemma is proved.
LEMMA 8. Suppose that G contains a normal subgroup X = X1 × · · · × Xs such that all subgroups
Xi , i = 1, . . . , s, are conjugate in G, and that G contains a subgroup H for which G = XH and X ∩ H = 1,
with NH (X1 ) = CH (X1 ). Also let h1 , . . . , hm be a full system of representatives of the right cosets of the
group H w.r.t. the subgroup NH (X1 ). Then m = s, and

CX (H) = {xh1 . . . xhs | x ∈ X1 }

is a full diagonal subgroup of X.

Proof. Clearly, the group H acts transitively on a set {X1 , . . . , Xs }. Therefore, for any j = 1, . . . , s,
ch h
there exists an h ∈ H such that Xi = X1h . Let h = chj , where c ∈ NH (X1 ). Then Xi = X1 j = X1 j . It is
h
not hard to note that X1hi = X1 j for 1  i < j  m. In this way m = s, and we may assume that Xi = X1hi
for any i = 1, . . . , s. Consider a set C = {xh1 . . . xhs | x ∈ X1 }. It is easy to see that C is a subgroup of

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X. Furthermore, C ≤ CX (H). Indeed, let h ∈ H, x ∈ X1 . We claim that (xh1 . . . xhs )h = xh1 . . . xhs . The
elements h1 h, . . . , hs h belong to different cosets of H w.r.t. NH (X1 ); therefore, for some permutation π on
a set {1, . . . , s} and for any i = 1, . . . , s, we have hi h ∈ NH (X1 )hiπ = CH (X1 )hiπ , that is, hi h = ci hiπ for
some element ci ∈ CH (X1 ). Thus

(xh1 . . . xhs )h = xh1 h . . . xhs h = xc1 h1π . . . xcs hsπ = xh1π . . . xhsπ = xh1 . . . xhs .

The inclusion C ≤ CX (H) is established.

We argue for the inverse. Let x ∈ CX (H). Then x = x1 . . . xs , where xi ∈ Xi , i = 1, . . . , s. Since x
commutes with each one of h1 , . . . , hs , for any i = 1, . . . , s we have

x1 . . . xs = x = xhi = xh1 i . . . xhs i .

It is worth observing that all elements xhj i , j = 1, . . . , s, belong to different groups X1 , . . . , Xs , and xh1 i ∈ Xi .
Equating the Xi th components of x1 . . . xs and xh1 i . . . xhs i , we obtain xi = xh1 i . Therefore x = xh1 1 . . . xh1 s .
Thus C = CX (H). It is easy to see that C is a full diagonal subgroup of X. The lemma is proved.
Remark. The requirement that NH (X1 ) = CH (X1 ) in the hypothesis of Lemma 8 is satisfied auto-
matically, provided that H is a transitive permutation group on {1, . . . , s} and G = X1  H, or if H is an
Abelian group of order s acting transitively by conjugations on X1 , . . . , Xs , since in the latter case NH (X1 )
is a point stabilizer, and
|NH (X1 )| = |H|/|H : NH (X1 )| = |H|/s = 1.

LEMMA 9. Assume that G contains a normal subgroup X = X1 × · · · × Xs such that Xi , i = 1, . . . , s,

are all conjugate in G. Also let H and H1 be cyclic subgroups of order s in G, so that H ∩ X = H1 ∩ X = 1
and G = HX = H1 X. Then H and H1 are conjugate in G.
Proof. It is easy to see that elements of the groups H and H1 form full systems of representatives of
the cosets of G w.r.t. the normal subgroup X. Let H = h. Since H G/X H1 , the unique element h1
of H1 lying in the coset hX generates H1 . It suffices to show that h and h1 are conjugates. Without loss of
i−1
generality we may assume that Xi = X1h , i = 1, . . . , s. Let h1 = hx1 . . . xs , where x1 ∈ X1 , . . . , xs ∈ Xs .
Then
s−1 s−1 s−1
1 = hs1 = (hx1 . . . xs ) . . . (hx1 . . . xs ) = hs (xh1 xh2 . . . xhs ) . . . (x1 . . . xs ) =
! "# $
s parentheses
s−1 s−2 s−1 s−2 s−1 s−2
(xh1 xh2 . . . xs ) (xh2 xh3 . . . x1 ) . . . (xhs xh1 . . . xs−1 ) .
! "# $! "# $ ! "# $
element Xs element X1 element Xs−1

This, in particular, implies

s−1 s−2
xh1 xh2 . . . xs = 1.

Let
s−2
u = (x−1 −1 −1 −1 −1 h −1 h
1 . . . xs−2 xs−1 )(x1 . . . xs−2 ) . . . (x1 ) .

A direct check shows that

s−1 s−2
hu1 = hxh1 xh2 . . . xs = h.

The lemma is proved.

LEMMA 10. Let s = pα , assume that G contains X = X1 × · · · × Xs as a normal subgroup, and
suppose that Xi , i = 1, . . . , s, are all conjugate in G. Assume also that H and H1 are elementary Abelian

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subgroups of order pα in G such that H ∩ X = H1 ∩ X = 1 and G = HX = H1 X. Then H and H1 are
conjugate in G.
The proof is by induction on α. We may assume that X = 1. H and H1 are elementary Abelian
subgroups acting regularly by conjugations on {X1 , . . . , Xs } by conjugations.
For α = 1, the statement follows from Lemma 9. Further we assume that α > 1. Let U be an hyperplane
in H, treated as an α-dimensional vector space over a field GF(p), and h be an element of H not lying in
U . The elements of H and H1 form full systems of representatives of the cosets of G w.r.t. the normal
subgroup X, and H G/X H1 ; therefore, the set U1 of elements of the group H1 lying in the cosets
whose representatives are elements of the group U is an hyperplane in H1 , and the element h1 ∈ H1 lying
in the coset hX does not belong to U1 .
It is easy to see that the orbits of U and U1 , and of V = h and V1 = h1 , coincide on {X1 , . . . , Xs }.
Put m = pα−1 . Let {Ω1 , . . . , Ωp } be a set of all orbits of U (and of U1 ) on {X1 , . . . , Xs } and {∆1 , . . . , ∆m }
be a set of all orbits of V (and of V1 ) on the same set. By Lemma 5, the set Ωi ∩ ∆j , 1  i  p, 1  j  m,
consists of just one element, which we denote by Xij . It is clear that

{X1 , . . . , Xs } = {Xij | 1  i  p, 1  j  m}.

Let Aj = X1j × · · · × Xpj . Then X = A1 × · · · × Am and the group XU = XU1 acts transitively by
conjugations on the set {A1 , . . . , Am }. By the inductive assumption, the subgroups U and U1 are conjugate
in XU . Therefore we may assume that U = U1 .
Let Bi = Xi1 × · · · × Xim . Since X = B1 × · · · × Bp and U normalizes each of the subgroups B1 , . . . , Bp ,
we have CX (U ) = C1 . . . Cp , where Ci = CBi (U ).
The subgroups V and V1 act transitively on {C1 , . . . , Cp }. By Lemma 1, C = CG (U ) = HCX (U ) =
H1 CX (U ). Consider a factor group C̄ = C/U . In this, the images of an element g ∈ C and of a subgroup
K ≤ C are denoted by ḡ and K̄, respectively. Let C0 = CX (U ). Then C̄ = V̄ C̄0 = V̄1 C̄0 and the groups
V̄ and V̄1 act transitively on {C̄1 , . . . , C̄p }. Furthermore, C̄0 = C̄1 × · · · × C̄p . By Lemma 9, V̄ and V̄1 are
conjugate in C̄. Consequently, H = U V and H1 = U V1 are conjugate in C. The lemma is proved.

3. PRIMITIVE SUPERLOCALS

The goal of the present section is to obtain a description of primitive superlocals in the group S(Ω). If
such a superlocal has a trivial radical, the superlocal itself coincides with S(Ω), and so below we assume
that N is a primitive superlocal of S(Ω), and P = Op (N ) = 1.
LEMMA 11. Let G be a natural semidirect product of an elementary Abelian p-group V of order pα ,
treated as an α-dimensional vector space over a field GF(p), and a group H = GL(V ). Suppose also that
Ω is the set of right cosets of G w.r.t. H, and that G acts by right shifts on Ω. Then the following hold:
(1) every right coset of G w.r.t. H contains exactly one element of V ;
(2) the subgroup V of G acts regularly on Ω;
(3) G acts faithfully on Ω.
(4) G acts on Ω doubly transitively and, consequently, primitively.
Proof. (1) Let g ∈ G, g = hv, where h ∈ H, v ∈ V . Then Hg = Hv; in particular, v ∈ Hg. Thus every
coset contains some element of V . Since the number of cosets coincides with the order of V , every coset
contains exactly one element of V .
(2) In this way the elements of V form a full system of representatives of the right cosets of G w.r.t.
H, and Ω = {Hv | v ∈ V }. Since every point Hv ∈ Ω, where v ∈ V , is obtained by shifting point H to

200
an element v, the subgroup V acts on Ω transitively. We have |V | = |Ω|; therefore, the stabilizer of every
point of Ω in V is trivial. This means that V acts on Ω regularly.
(3) Let g ∈ G stabilize every element of Ω. Assume g = hu, where h ∈ H and u ∈ V . Then, in
particular,
Hu = Hhu = Hg = H.

Hence u stabilizes point H in Ω. By (2), u = 1 and g ∈ H. This means that the equality Hvg = Hv holds
for any v ∈ V . At the same time, Hvg = Hgg −1 vg = Hv g , with v g ∈ V . In view of (1), v g = v. Hence
g ∈ H, treated as a linear transformation of the space V , acts trivially on V . Consequently, g = 1.
(4) It suffices to prove that the stabilizer in G of some point from Ω acts transitively on a set of the
remaining points. The stabilizer of a point H ∈ Ω is the subgroup H itself. Therefore we need only show
that H acts transitively on Ω \ {H} = {Hv | v ∈ V # }. If h ∈ H, then Hvh = Hv h , and since H = GL(V )
acts transitively on V # by conjugations, the result follows immediately. The lemma is proved.
A consequence of Lemmas 4 and 11 is the following:
THEOREM 4. Let G = S(Ω). A primitive subgroup N of G such that Op (N ) = 1 is a superlocal
in G if and only if |Ω| = pα for some natural number α and N is the normalizer in G of some transitive
elementary Abelian subgroup of G. Moreover, N is isomorphic to a natural semidirect product of the
elementary Abelian p-group Epα , treated as an α-dimensional vector space over a field GF(p), and the
group GL(Epα ) isomorphic to GLα (p). Such superlocals (resp., their radicals) are all conjugate in G.
Proof. Let |Ω| = pα and P be some transitive elementary Abelian subgroup of G. By Lemma 4,
all transitive elementary Abelian subgroups of G are conjugate, and Lemma 11 implies that N = NG (P )
contains a subgroup isomorphic to a natural semidirect product of the group Epα , whose image coincides
with P , and the group GL(Epα ). On the other hand, N/P = NG (P )/CG (P ) by Lemma 4, and so N/P is
isomorphically embeddable in the group AutEpα = GLα (p). Hence N is a split extension of Epα by GLα (p).
In particular, Op (N ) = P , that is, N is a primitive superlocal in G.
Conversely, let N be a primitive superlocal of G and P = Op (N ) = 1. Since Φ(P )  N , the subgroup
Φ(P ) should be either transitive or trivial. The former case is ruled out by Lemma 7. This means that
Φ(P ) = 1 and P is a transitive elementary Abelian subgroup of G.
Such superlocals are all conjugate in view of Lemma 4. The theorem is proved.

4. SUPERLOCALS WITH TRANSITIVE RADICALS

The objective of the present section is to obtain a description of superlocals in a group S(Ω) with
transitive radicals. If S(Ω) possesses such a superlocal then the number |Ω| ought to divide the order of a
radical; consequently, it must be the power of a prime p.
LEMMA 12. Let G = S(Ω), where |Ω| = pα , and P be a transitive elementary wreath product of type
(α1 , . . . , αk ) in G. Then the following statements hold:
(1) The group P acts transitively on Ω.
(2) If k > 1 and groups E, P0 , P1 , . . . , Ppαk and sets Ω1 , . . . , Ωpαk are as in the definition of a transitive
elementary wreath product, then Φ(P ) ≤ P0 and the coordinate projection in P0 of the group Φ(P ) onto
Pi coincides with Pi , i = 1, . . . , pαk . In particular, ∆ = {Ω1 , . . . , Ωpαk } is the set of orbits of Φ(P ).
(3) A transitive elementary wreath product has a uniquely determined type.
(4) If Q is a transitive elementary wreath product of type (β1 , . . . , βl ) then subgroups P and Q are
conjugate in G iff (α1 , . . . , αk ) = (β1 , . . . , βl ).

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 (5) The normalizer in G of a subgroup P is a split extension of the group P by the group GLα1 (p) ×
· · · × GLαk (p).
(6) The factor group NG (P )/P is equal to 1 iff p = 2, k = α, and α1 = · · · = αk = 1, that is, if P is a
Sylow 2-subgroup of G.
Proof. (1) Is proved readily using induction on k.
(2) Follows immediately from Lemma 7.
(3) Follows by induction from (2).
(4) The necessity follows from (3) in the obvious way. Let Q be a transitive elementary wreath product
of type (α1 , . . . , αk ). If k = 1, then P and Q are transitive elementary Abelian subgroups of G, and so
these are conjugate in G. Let k > 1. There is no loss of generality in assuming that the sets Ω1 , . . . , Ωpαk
and the group S via which P and Q are defined are same. Let P1 and Q1 be transitive elementary wreath
products of type (α1 , . . . , αk−1 ) in the group S(Ω1 ), used to define P and Q, respectively. Assume that E
and F are transitive elementary Abelian subgroups of S such that
 g  g
P =E P1 , Q = F Q1 .
g∈E g∈F

Let {P1 , . . . , Ppαk } ={P1g | g ∈ E}, {Q1 , . . . , Qpαk } = {Qg1 | g ∈ F }, Pi , Qi ≤ S(Ωi ), i = 1, . . . , pαk ,
P0 = P1 × · · · × Ppαk , Q0 = Q1 × · · · × Qpαk , Xi = S(Ωi ), i = 1, . . . , pαk . The subgroups P and Q are,
therefore, contained in a subgroup H = SX1 . . . Xpαk . By the inductive assumption, Pi and Qi are conjugate
in Xi ; so, if we replace Q by a suitable conjugate subgroup we may assume that Pi = Qi , i = 1, . . . , pαk ,
and P0 = Q0 . Truly, the statement that F ≤ S may turn out to be invalid in this case, but P, Q ≤ H
anyway.
Consider a group NH (P0 ) = SN1 . . . Npαk , where Ni = NXi (Pi ). It is clear that E, F ≤ NH (P0 ) and
E ∩ N0 = F ∩ N0 = 1, where N0 = N1 × · · · × Npαk  NH (P0 ). Since NH (P0 ) acts on Γ = {N1 , . . . , Npαk } by
conjugations, the corresponding homomorphism π : NH (P0 ) → S(Γ) is determined, and ker π = N0 . The
images of subgroups E and F in S(Γ) are transitive elementary Abelian subgroups; consequently, these are
conjugate in S(Γ). We may assume that EN0 = F N0 . Then E and F are conjugate in EN0 = F N0 by
Lemma 10. This means that P and Q are also conjugates.
(5) Is proved by induction on k. For k = 1, P is a transitive elementary Abelian subgroup, and
N = NG (P ) is a split extension of P by GLα (p) by Theorem 4. Let k > 1 and sets Ω1 , . . . , Ωpαk and
groups S, E, P0 , P1 , . . . , Ppαk be as in the definition of a transitive elementary wreath product. Since the
sets Ω1 , . . . , Ωpαk are orbits of the characteristic subgroup Φ(P ) of P , these form an imprimitivity system for
N = NG (P ). Let Xi = S(Ωi ), i = 1, . . . , pαk . The group N normalizes a subgroup P0 = P ∩ (X1 . . . Xpαk ),
and so N ≤ NG (P0 ) = SN0 , where N0 = NX1 (P1 ) × · · · × NXpαk (Ppαk ). For brevity, we write K for a
subgroup NG (P0 ) and consider the group K̄ = K/P0 .
The images of an element g ∈ K and of a subgroup H ≤ K in the factor group K̄ are denoted by ḡ and
H̄, respectively. It is easy to see that N is a preimage in K of the group NK̄ (P̄ ). Since P̄ ∩ N̄0 = 1, P̄ ≤ S̄,
and N̄0 = N̄1 × · · · × N̄pαk , where Ni = NXi (Pi ), in view of Lemmas 1 and 8, NK̄ (P̄ ) = CN̄0 (P̄ )NS̄ (P̄ ) and
CN̄0 (P̄ ) = CN̄0 (S̄) is a full diagonal subgroup in the direct product N̄0 = N̄1 × · · · × N̄pαk . Therefore

CN̄0 (P̄ ) N̄1 = N1 P0 /P0 N1 /(N1 ∩ P0 ) = N1 /P1 GLα1 (p) × · · · × GLαk−1 (p)

by the inductive assumption. For NS̄ (P̄ ) centralizes CN̄0 (P̄ ) (which follows from Lemma 8), we are given
the isomorphism
NK̄ (P̄ ) GLα1 (p) × · · · × GLαk−1 (p) × NS̄ (P̄ ).

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By Theorem 4, NS̄ (P̄ ) is a split extension of P̄ by GLαk (p). Hence

NG (P )/P NK̄ (P̄ )/P̄ GLα1 (p) × · · · × GLαk−1 (p) × GLαk (p).

And it is easy to see that the extension of P by NG (P )/P is split.

(6) Follows immediately from (5). The lemma is proved.
THEOREM 5. Transitive elementary wreath products and no other are transitive radicals in the
group G = S(Ω), where |Ω| = pα .
Proof. The fact that transitive elementary wreath products are transitive radicals in G follows from
Lemma 12. We argue for the way back. A proof is by induction on α.
For α = 1, every non-trivial p-subgroup of G is a transitive cyclic group of order p; consequently, it is a
transitive elementary wreath product of type (1).
Let α > 1. Assume that P is a transitive radical in G. By Lemma 6, the Frattini subgroup Φ(P ) of P is
not transitive, while the groups P and N = NG (P ) act transitively on the set ∆ = {Ω1 , . . . , Ωs } of orbits of
Φ(P ). Let s = pβ . Since s > 1, we can say that β
1. In this instance |Ωi | = pα−β for any i = 1, . . . , s. If
|Ωi | = 1 then Φ(P ) is trivial, in which case P is a transitive elementary Abelian subgroup of G. Therefore
we may assume that |Ωi | = pα−β > 1.
The group NG (P ) normalizes a subgroup X = X1 × · · · × Xs , where Xi = S(Ωi ), i = 1, . . . , s. In virtue
of Propositions 4 and 5, P0 = P ∩ X = P1 × · · · × Ps , where Pi = P ∩ Xi . In view of P being transitive,
all subgroups Pi are conjugate in P . By the inductive assumption, Pi is a transitive elementary wreath
product of some type (α1 , . . . , αk ) in the group Xi . Since all wreath products of equal type are conjugate
in Xi , for any i = 1, . . . , s, there exists a permutation gi = (x11 xi1 ) . . . (x1pα−β xipα−β ) ∈ G such that Ωl =
{xlj | 1  j  pα−β } and Pi = P1gi . In fact, there are permutations hi = (x11 yi1 ) . . . (x1pα−β yipα−β ) ∈ G
and  
yi1 . . . yipα−β
fi = ∈ Xi
xi1 . . . xipα−β

for which Ωi = {yij | 1  j  pα−β }, Gi = Gg1i , and P1hi fi = Pi . As gi we can take fi−1 hi fi . Let
S = gi | i = 2, . . . , pβ . Then S Spβ . The group S acts on ∆ = {Ω1 , . . . , Ωpβ } and can be identified with
S(∆) the obvious way. Letting E be a transitive elementary Abelian subgroup of S, we may specify the
transitive elementary wreath product  g
Q=E P1 .
g∈E

Now it suffices to show that P and Q are conjugate in G.

Let Y = SX = NG (X). Since NG (P ) ≤ Y and NG (P ) normalizes P0 , the inclusion NG (P ) ≤ NY (P0 )
holds. If we apply an argument similar to Frattini’s we may conclude that Y = XNY (P0 ), whence

NY (P0 )/NX (P0 ) XNY (P0 )/X = Y /X S.

Since S ≤ NY (P0 ) and S ∩ X = 1, the equality NY (P0 ) = NX (P0 )S holds. Furthermore, N0 = NX (P0 ) =
N1 × · · · × Npβ , where Ni = NXi (Pi ). The group P acts transitively on Γ = {N1 , . . . , Npβ }, and so its image
in S(Γ), under the natural homomorphism σ : Y → S(Γ), is a transitive group. Moreover, since ker σ = N0 ,
P ∩ N0 = P0 , and Φ(P ) ≤ P0 , this image, isomorphic to P/P0 , is a transitive elementary Abelian subgroup
of S(Γ).
Since the images of P and E in S(Γ) are conjugates, we may assume that P N0 = EN0 = T . Lemma 12
implies that Op (N0 ) = P0 , and so the subgroup P0 is normal in T . Let T̄ = T /P0 . If g ∈ T and H ≤ T

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then by ḡ and H̄ we denote the images of, respectively, an element g and a subgroup H in T /P0 . We have
T̄ = P̄ N̄0 = Ē N̄0 , in which case P̄ ∩ N̄0 = Ē ∩ N̄0 = 1, the group T̄ acts transitively on {N̄1 , . . . , N̄pβ }, and
N̄0 = N̄1 × · · · × N̄pβ . By Lemma 10, the subgroups P̄ and Ē are conjugate in T̄ , and consequently their
preimages, that is, subgroups P and EP0 = Q, are also conjugate in T . The theorem is proved.

5. PROOF OF THEOREM 1

Let a subgroup P of G be an mth independent degree of a transitive elementary wreath product P1 of

type (α1 , . . . , αk ) in S(Ω1 ), and |Ω| = mpα1 +···+αk = |Ω1 |m. Then there exists a partition Ω = Ω1 ∪. . .∪Ωm
of the set Ω into a union of mutually disjoint sets Ωi , i = 1, . . . , m, of equal power s = pα1 +···+αk , which
are orbits of P on Ω. By the definition of an mth independent degree, there is a set {g1 , g2 , . . . , gm } ⊆ G
such that Ωi = Ω1 g1 and P = P1g1 . . . P1gm . As in the proof of Theorem 5, we may assume that g1 = 1,
gi = (x11 xi1 ) . . . (x1s xis ) for i = 2, . . . , m, where Ω1 = {x11 , . . . , x1s } and Ωi = {xi1 , . . . , xis }.
Let Pi = P1gi , i = 1, . . . , m; N = NG (P ); X = X1 × · · · × Xm , where Xi = S(Ωi ). Since P is normal in
N , the group N acts on the set {Ω1 , . . . , Ωm } of orbits of P ; consequently, N normalizes the subgroup X.
Let Y = NG (X). Then Y /X Sm . In virtue of the fact that elementary wreath products of equal type
are conjugate in X1 = S(Ω1 ), it is not hard to see that, for any y ∈ Y , there exists an element x ∈ X such
that P y = P x . Using the Frattini argument, it is therefore easy to show that Y = NY (P )X. Hence

NY (P )/NX (P ) = NY (P )/(X ∩ NY (P )) NY (P )X/X = Y /X Sm .

Let S = gi | i = 1, . . . , m. Clearly, S ∩ X = 1, S Sm , and S ≤ NG (X) = Y . Therefore Y = XS.

It is also clear that the subgroup S normalizes P and NX (P ). This implies N = NY (P ) = NX (P )S by
Lemma 1. Furthermore, it is obvious that

NX (P ) = NX1 (P1 ) . . . NXm (Pm ),

Op (NX (P )) = Op (NX1 (P1 )) . . . Op (NXm (Pm )) = P1 . . . Pm = P.

Assume Q = Op (N ) = P . In view of Lemmas 2 and 3, one of the following cases holds:
(1) p = 2 and m = 2;
(2) p = 3 and m = 3;
(3) p = 2 and m = 4.
Consider a factor group N/P and its subgroup QNX (P )/P . By Lemma 3, the group Q/P is elementary
Abelian and acts transitively by conjugations on the set {NX1 (P1 )P/P, . . . , NXm (Pm )P/P }. On the other
hand, since Q  N , NX (P )  N , and Q ∩ NX (P ) = P , we have [NX (P )/P, Q/P ] = 1. Consequently,
NX (P )/P = 1 and NX1 (P1 )/P1 NX1 (P1 )P/P ≤ NX (P )/P = 1. By Lemma 12, this is possible only if
p = 2, α1 = · · · = αk = 1, and m equals 2 or 4, which clashes with the hypothesis of the theorem. Hence
Op (N ) = P , that is, N is a superlocal in G.
Conversely, let P be a non-trivial radical of some transitive superlocal N . Then N acts on the set
{Ω1 , . . . , Ωm } of orbits of P . In particular,

|Ω1 | = · · · = |Ωm | = |Ω|/m = pα

for some α
1. The group N normalizes a subgroup X = X1 × · · · × Xm , where Xi = S(Ωi ), i = 1, . . . , m.

In this event P ≤ X. By Proposition 5, therefore, P is a direct product of subgroups Pi = P ∩ Xi ,
i = 1, . . . , m, and each one of Pi is a non-trivial transitive radical in Xi = S(Ωi ).

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 By Theorem 5, Pi is a transitive elementary wreath product in Xi . Since N acts on {Ω1 , . . . , Ωm }
transitively, for any i = 2, . . . , m, there exists an element gi ∈ N such that Ωi = Ω1 gi . It follows that
Xi = X1gi and Pi = P ∩ Xi = P ∩ X1gi = (P ∩ X1 )gi = P1gi . Hence P is an mth independent degree of the
transitive elementary wreath product P1 of the group X1 ; here, |Ω| = m|Ω1 | = mpα = mpα1 +···+αk , where
(α1 , . . . , αk ) is the type of P1 . If p = 2, α1 = · · · = αk , and m equals 2 or 4 then N P1  S2 or N P1  S4 .
In either event, Op (N ) = P , which contradicts the assumption that P is a radical of the superlocal N . The
theorem is proved.

6. PROOF OF THEOREM 2

Let Ω = Ω1 ∪ · · · ∪ Ωm , where Ωi = ∅ and Ωi ∩ Ωj = ∅ for i = j. Assume that Ni is a transitive

superlocal in a group Xi = S(Ωi ), i = 1, . . . , m, such that, among Pi = Op (Ni ), there is at most one
trivial group, and transitive elementary wreath products whose independent degrees are non-trivial groups
Pi are of different types for distinct i. Let {∆i1 , . . . , ∆iki } be the set of orbits of Pi , i = 1, . . . , m. Suppose
P = P1 . . . Pm and N = NG (P ). Then Γ = {∆is | 1  i  m, 1  s  ki } is the set of orbits of P . We
show that N = N1 . . . Nm , thereby proving that N is a superlocal. The definition of P implies that, for
any 1  i  m and 1  s  ki , either P ∩ S(∆is ) = Pi ∩ S(∆is ) is a transitive elementary wreath product
in the group S(∆is ), or |∆is | = 1. We claim that {Ω1 , . . . , Ωm } is the set of orbits of N . The group N acts
on Γ. Assume that g ∈ N is such that ∆is g = ∆jt for some 1  i, j  m, 1  s  ki , and 1  t  kj . Then
S(∆is )g = S(∆jt ) and (P ∩ S(∆is ))g = P ∩ S(∆jt ). In particular, either P ∩ S(∆is ) = P ∩ S(∆jt ) = 1,
or the transitive elementary wreath products P ∩ S(∆is ) and P ∩ S(∆jt ) are of equal type. Thus i = j.
It follows that every orbit of N is contained in one of the sets Ωi , 1  i  m. On the other hand,
N1 . . . Nm ≤ NG (P ) = N , and consequently each set Ωi will be contained in some orbit of N by the choice
of N1 , . . . , Nm . In this way {Ω1 , . . . , Ωm } is the set of orbits of N . Therefore N ≤ X1 . . . Xm . Let g ∈ N
and g = g1 . . . gm , where gi ∈ Xi , 1  i  m. It is easy to see that gi ∈ NXi (Pi ) = Ni . This implies
N ≤ N1 . . . Nm . Since the inverse inclusion holds also, N is a superlocal in G.
Conversely, let N be a superlocal in G, P = Op (N ), and {Ω1 , . . . , Ωm } be the set of orbits of N . Then
N ≤ X1 × · · · × Xm , where Xi = S(Ωi ), and by Proposition 5, N = N1 . . . Nm , where Ni is a transitive
superlocal in Xi , 1  i  m. Let Pi = Op (Ni ). Assume that, for some distinct i and j, groups Pi
and Pj either both are trivial or both are independent degrees of transitive elementary wreath products
of equal type. In the former case, Ni = Xi and Nj = Xj , and so S(Ωi ∪ Ωj ) normalizes P , that is,
S(Ωi ∪ Ωj ) ≤ NG (P ) = N . We have arrived at a contradiction with the fact that Ωi and Ωj are orbits of N .
In the latter case, the group Pi Pj is an independent degree of some transitive elementary wreath product.
It is easy to see that the group NS(Ωi ∪Ωj ) (Pi Pj ) acts transitively on Ωi ∪ Ωj , and is contained in N , which
is a contradiction with Ωi and Ωj being orbits of N . The theorem is proved.

7. PROOF OF THEOREM 3

We make use of Proposition 4.

(1) Let NA be a superlocal in the group A and PA = Op (NA ). By Proposition 4, there exists a superlocal
NS = NS (PA ) in S such that NA = A ∩ NS . Assume PS = Op (NS ) ≤ A. Then PA = A ∩ PS < PS and
Op (NS (PA )) = Op (NS ) = PA , and so the group PA = A ∩ Op (NS ) is not a radical in S. Conversely, let NS
be a superlocal in S, and PS = Op (NS ). If PS ≤ A then PS = PS ∩ A is a radical, and NS ∩ A = NA (PS )
is a superlocal, in A. If, however, PA = PS ∩ A is not a radical in S then |PS : PA | = 2. Clearly,

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P = Op (NS (PA )) > PA and |P : PA | = 2. Moreover, the group NS normalizes Op (NS (PA )) = P , and so
P ≤ PS and P = PS . In view of Proposition 4,

NA = NS ∩ A = NS (P ) ∩ A = NS (PA ) ∩ A = NA (PA )

is a superlocal in A.
(2) Follows from Proposition 4.
Acknowledgement. I express my deep gratitude to R. Zh. Aleev and V. D. Mazurov for granting me
information about the history of the problem in hand, and to V. M and E. N. Arkhipovskii for helping me
prepare the paper for publication.

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