Free abelian groups
Definition 2.19 Let F be an abelian group (written
additively). A set X ⊂ F is said to be a basis of F
if F = hX i and for any positive integer n, if
x1 , . . . , xn are distinct in X , then
α1 x1 + · · · + αn xn = 0
if and only if α1 = · · · = αn = 0.
Theorem 2.20 Let F be an abelian group. The
following statements are equivalent:
(i) F has a nonempty basis
(ii) F is a direct sum of a family of infinite cyclic
group.
(iii) F is a free object in the category of abelian
groups.
Theorem 2.21 Any two bases of a free abelian
group F have the same cardinality.
Theorem 2.22 Suppose X1 and X2 are bases of F1
and F2 respectively. If |X1 | = |X2 |, then F1 ∼
= F2 .
Theorem 2.23 Let F be a free abelian group of
finite rank n and G a nonzero subgroup of F . There
exist a basis {x1 , x2 , . . . , xn } and positive integer
d1 , . . . dt such that
d1 |d2 |d3 · · · dt−1 |dt
and {d1 x1 , . . . , dt xt } form a basis of G .
Corollary 2.24 Suppose G is an abelian group
generated by n elements. Then for any subgroup H
of G , H can be generated by m elements with
m ≤ n.
Theorem 2.25 Let G be a finite generated abelian
group G . Then
G∼
= F ⊕ Zd1 ⊕ · · · ⊕ Zdr .
such that F is free (F could be empty when G is
finite) and d1 |d2 | · · · |dr .
If we define Tor (G ) = {g ∈ G : ◦(g ) is finite}. It is
easy to see that Tor (G ) is a subgroup and
Tor (G ) ∼
= Zd1 ⊕ · · · ⊕ Zdr
in Theorem 2.24.
Corollary 2.26 Let G be a finite generated abelian
group G .
(i) G is free if Tor (G ) = {0}.
(ii) G /Tor (G ) is free.
(iii) If G is not finite, then there exists a free group
F in G such that G = F ⊕ Tor (G ).
Theorem 2.27 Let G and G 0 be finitely generated
abelian group. G and G 0 is isomorphic if and only if
Tor (G ) ∼
= Tor (G 0 ) and G /Tor (G ) ∼
= G 0 /Tor (G 0 ).
An abelian group G is said to be divisible if for any
g ∈ G , any positive integer n, there exists h such
that nh = g .
Theorem 2.28 A divisible group is a direct
summand of every abelian group, i.e. for any
H < G , there exists N < G such that N + H = G
and N ∩ H = {0}.
Let G be a group. Let C1 (G ) = Z (G ). Generally,
for any positive integer i with Ci (G )4G . We define
Ci+1 (G ) to be the ηi−1 (Z (G /Ci (G )) where
ηi : G → G /Ci (G ) is the canonical mapping. We
thus get an ascending series
{e} < Z (G ) = C1 (G ) < C2 (G ) < · · ·
Definition 2.29 A group G is nilpotent if
Cn (G ) = G for some n.
Example. Any abelian group is nilpotent.
Theorem 2.30 Every finite p-group is nilpotent.
Theorem 2.31 Suppose G1 , . . . , Gn are nilpotent.
Then G1 × G2 · · · × Gn is also nilpotent.
Lemma 2.32 Let G be nilpotent and H < G . Then
H ( N(H).
Theorem 2.33 A finite group G is nilpotent if and
only if G is a direct product of its Sylow subgroups.
Theorem 2.34 Let G be a finite nilpotent group.
Suppose m divides |G |. Then there exists H < G
with |H| = m. Moreover, G is solvable.
Definition 2.35 A subgroup H < G is said to be a
characteristic [resp. fully invariant] subgroup of G if
for any automorphism [resp. endomorphism]
σ : G → G , σ(H) = H.
Note (i) Any fully invariant subgroup is a
characteristic subgroup and any characteristic
subgroup is normal.
(ii) Any normal Sylow subgroup of G is fully
invariant.
Theorem 2.36 If G is solvable and N and is a
minimal normal subgroup in G , then N is an abelian
p-group for some prime p.
Theorem 2.37 Let G be a finite solvable group of
order mn, with (m, n) = 1. Then G contains a
subgroup of order m. Moreover, any two subgroups
or order m are conjugate; and any subgroup of order
k|m is contained in a subgroup of order m.
Inverse Limit
Let {Gi }i∈N be a sequence of groups. For each
i ≥ 2, there exists surjective homomorphisms
fi : Gi → Gi−1 .
Let
−(Gn , fn ) = {(x1 , x2 , . . .) : ∀i ≥ 2, fi (xi ) = xi−1 }.
lim
←
Theorem 2.38 lim←−(Gn , fn ) is a group under
operation ∗ such that
(x1 , x2 , . . .) ∗ (y1 , y2 , . . .) = (x1 y1 , x2 y2 , . . .).
Completion
Suppose G is a group and there is a sequence of
normal subgroups {Hi } with Hi ⊃ Hi+1 for all i.
A sequence {xi } in G is said to be Cauchy sequence
if given Hr , there exists N such that for all
m, n ≥ N, xn xm−1 ∈ Hr .
We say {xi } is a null sequence if given Hr , there
exists N such that for all n ≥ N, xn ∈ Hr .
Let Ĝ be the set of all Cauchy sequence, in Ĝ , we
define {xi } · {yi } = {xi yi }. Let G n be the set of all
null sequence.
Theorem 2.39 Ĝ is a group under · and G n is a
normal subgroup of Ĝ . Moreover,
Ĝ /G n ∼
= lim
← − G /Hr .
