Summary Notes

1 Order of an Element Theorem 2.5. Let ord(g) = n, and let k ∈ Z+ . Then

⟨g k ⟩ = ⟨g gcd(n,k) ⟩.
Definition 1.1 (Order of Element). Let G be a group Addition notation: ⟨kg⟩ = ⟨gcd(n, k)g⟩.
and e ∈ G is the identity element. The order of an
element g ∈ G is the smallest positive integer n such Corollary 2.6 (Orders of Elements in Finite Cyclic
that g n = e. Groups). In a finite cyclic group, the order of an ele-
ment divides the order of the group.
Remark 1.2. If no such integer exist, then g is said
Corollary 2.7 (Criterion for ⟨ai ⟩ = ⟨aj ⟩ and
to have infinite order. The order of an element g is
|ai | = |aj |). Let ord(g) = n. Then ⟨g i ⟩ = ⟨g j ⟩ if and
denoted by ord(g) or |g|.
only if gcd(n, i) = gcd(n, j) and ord(g i ) = ord(g j ) if
Theorem 1.3. Let G be a group and g ∈ G. Then the and only if gcd(n, i) = gcd(n, j).
following hold
Corollary 2.8 (Generators of Finite Cyclic Groups).
• ord(g) = ord(g ).
−1 Let ord(g) = n. Then ⟨g⟩ = ⟨g j ⟩ if and only if
gcd(n, j) = 1 and ord(g) = |⟨g i ⟩| if and only if
• If ord(g) = n and g = e, then n|m.
m
gcd(n, j) = 1.
• If ord(g) = n then ord(g m ) = n
gcd(n,m)
. Corollary 2.9 (Generators of Zn ). An integer k in Zn
is a generator of Zn if and only if gcd(n, k) = 1.
Theorem 2.10. Let G be a cyclic group with genera-
2 Cyclic Group tor g.
Definition 2.1 (Cyclic Group). A group G is called 1. If the order of G is infinite then G is isomorphic
cyclic if there exists g ∈ G such that G = {g n | n ∈ Z}. to Z.
The element g is called the generator for G. The
cyclic group G with generator g will be denoted by 2. If the order of G is finite then G is isomorphic to
⟨g⟩ = G. Zn .

NOTE: For addition notation, we have G = {ng|n ∈ Recall: Isomorphism is a group homomorphism
Z}. and a bijective map (one-to-one and onto).

The generator of a cyclic group is not unique. Theorem 2.11 (Fundamental Theorem of Cyclic
Groups). Let G = ⟨g⟩ be a cyclic group.
Theorem 2.2 (Infinite and Finite Cyclic Group). Let 1. Every subgroup of a cyclic group is cyclic.
G be a group, and let g belong to G.
2. If |⟨g⟩| = n, then the order of any subgroup of ⟨g⟩
1. If g has infinite order, then g i = g j if and only if is a divisor of n.
i = j.
3. For each positive k, which is a divisor of n, the
2. If g has a finite order, say, n, then ⟨g⟩ = group ⟨g⟩ has exactly one subgroup of the order k,
{g 0 , g 1 , g 2 , . . . , g n−1 } and g i = g j if and only if n
namely ⟨g k ⟩.
n divides i − j.
Corollary 2.12 (Subgroups of Zn ). For each positive
divisor k of n, the set ⟨ nk ⟩ is the unique subgroup of Zn
Every group has a subgroup which is cyclic
of order k. Moreover, these are the only subgroups of
Zn .
Corollary 2.3. ord(g) = |⟨g⟩|.
Definition 2.13 (Euler Phi Function). ϕ(n) is the
Corollary 2.4. If a and b belong ti a finite group and number of positive integers less than n > 1 and rela-
ab = ba, then ord(ab) divides ord(a) · ord(b) tively prime to n. ϕ(n) = |U (n)|.
Theorem 2.14. Let d be a positive divisor on n. The Theorem 3.7 (Product of 2-Cycles). Every permuta-
number of elements of order d in a cyclic group of tion in Sn , n > 1, is a product of 2-cycles.
order n is ϕ(d).
Lemma 3.8. If ϵ = β1 β2 · · · βr , where the β’s are 2-
Corollary 2.15. In a finite group, the number of ele- cycles, then r is even.
ments of order d is a multiple of ϕ(d).
Theorem 3.9 (Always Even or Always Odd). If a
Definition 2.16 (Property of ϕ(n)). For a prime p, permutation α can be expressed as a product of an even
(odd) number of 2-cycles, then every decomposition of
• ϕ(p) = p − 1. α into a product of 2-cycles must have an even(odd)
• ϕ(p2 ) = p2 − p = p(p − 1). number of 2-cycles. In symbols, if

• ϕ(pn ) = pn−1 (p − 1). α = β1 β2 · · · βr and α = γ1 γ2 · · · γs ,

• gcd(m, n) = 1, ϕ(mn) = ϕ(m)ϕ(n). where the β’s and the γ’s are 2-cycles, then r and s
are both even or both odd.
Definition 2.17 (Subgroup Lattice). This is to rep-
resent the relationship among subgroups of a group. Definition 3.10 (Even and Odd Permutations). A
permutation that can be expressed as a product of an
• The lattice contains ALL subgroups of G. even number of 2-cycles is called an even permutation.
A permutation that can be expressed as a product of an
• Let H, K be subgroups of G. H is connected to K
odd number of 2-cycles is called an odd permutation.
at a higher level if and only if H ⊂ K.
Theorem 3.11 (Even Permutations Form a Group).
The role of cyclic groups is to serve as building The set of even permutations in Sn forms a subgroup
blocks for all finite Abelian groups in much the of Sn .
same way that primes are the building blocks for
the integers. Definition 3.12 (Alternating Group of Degree n).
The group of even permutations of n symbols is de-
noted by An and is called the alternating group of de-
gree n.
3 Permutation Groups Theorem 3.13. For n > 1, An has order n!/2.
Definition 3.1 (Permutations of A, Permutation
Group of A). A permutation of a set A is a function 4 Cosets and Lagrange’s Theo-
from A to A that is both one-to-one and onto. A per-
mutation group of a set A is a set of permutations of rem
A tha forms a group under function composition.
Definition 4.1 (Coset of H in G). Let G be a group
Definition 3.2 (Symmetric Group, Sn ). Let A = and let H be a nonempty subset of G. For any a ∈ G,
{1, 2, . . . , n}. The set of all permutations of A is called the set {ah| h ∈ H} is denoted by aH. Analogously,
the symmetric group of degree n and is denoted Sn . Sn Ha = {ha| h ∈ H} and aHa−1 = {aha−1 | h ∈ H}.
has n(n − 1) = n! elements. When H is a subgroup of G, the set aH is called the
Definition 3.3 (Cycle Notation). Another notation left coset of H in G cotaining a, whereas Ha is called
commonly used to specify notations is called cycle no- the right coset of H in G containing a. In this case, the
tation. An expression of the form (a1 a2 . . . am ) is called element a is called the coset representative of aH (or
a cycle of length m or an m-cycle. Ha). We use |aH| to denote the number of elements in
the set aH, and |Ha| to denote the number of elements
Theorem 3.4 (Products of Disjoint Cycles). Every in Ha.
permutation of a finite set can be written as a cycle or
Lemma 4.2 (Properties of Cosets). Let H be a sub-
as a product of disjoint cycle.
group of G, and let a and b belong to G. Then,
Theorem 3.5 (Disjoint Cycles Commute). If the pair
1. a ∈ aH.
of cycles α = {a1 , a2 , . . . , am } and β = {b1 , b2 , . . . , bn }
have no entries in common, then αβ = βα. 2. aH = H if and only if a ∈ H.
Theorem 3.6 (Order of a Permutation). The order of 3. (ab)H = a(bH) and H(ab) = (Ha)b.
a permutation of a finite set written in disjoint cycle
form is the least common multiple of the lengths of the 4. aH = bH if and only if a ∈ bH.
cycles. 5. aH = bH or aH ∩ bH = ∅.
 6. aH = bH if and only if a−1 b ∈ H. 3. gH = Hg ∀g ∈ G [gh1 = h2 g].
7. |aH| = |bH|.
If H ◁ G, then G/H = {xH|x ∈ G} denotes
8. aH = Ha if and only if H = aHa−1 . the set of left cosets of H in G.
9. aH is a subgroup of G if and only if a ∈ H.
Theorem 4.14. Let H be a subgroup of a group G.
Then, left coset multiplication is well-defined by the
Indeed, we may view the cosets of H as a par-
equation (aH)(bH) = (ab)H if and only if H is a nor-
titioning of G into equivalence classes under
mal subgroup of G.
the equivalence relation defined by a ∼ b if
aH = bH. Corollary 4.15. Let H be a normal subgroup of G.
Then, the cosets of H form a group G?H under the
Theorem 4.3 (Lagrange’s Theorem: |H| Divides binary operation (aH)(bH) = (ab)H.
|G|). If G is a finite group and H is a subgroup of G, Definition 4.16 (Quotient Group or Factor Group).
then |H| divides |G|. Moreover, the number of distinct The group G/H is called the quotient group or factor
left (right) cosets of H in G is |G|/|H|. group of G by H.
Corollary 4.4 (|G : H| = |G|/|H|). If G is a fi-
nite group and H is a subgroup of G, then |G : H| = |G/H|= [G:H]
|G|/|H|.
Corollary 4.5 (|a| Divides |G|). In a finite group, the
order of each element of the group divides the order of
the group.
Corollary 4.6 (Groups of Prime Order Are Cyclic).
A group of prime order is cyclic.
Corollary 4.7 (a|G| = e). Let G be a finite group, and
let a ∈ G. Then, a|G| = e.
Theorem 4.8 (|HK| = |H||K|/|H ∩ K|). For two
finite subgroups H and K of a group, define the
set HK = {hk|h ∈ H, k ∈ K}. Then |HK| =
|H||K|/|H ∩ K|.
Definition 4.9 (The Index of a Subgroup). Let H be
a subgroup of a group G. The number of distinct right
(or left) cosets of H in G is called the index of H in
G, denoted by (G : H) or |G : H|.
Theorem 4.10. Suppose H and K are subgroups of a
group GG such that K ≤ H ≤ G and suppose (H : K)
and (G : H) are both finite. Then, (G : K) is finite
and (G : K) = (G : H)(H : K).
Remark 4.11. It is important to note that (G : H) =
|G|
|H|
.

Definition 4.12 (Normal Subgroup). A subgroup H

of a group G is normal if its left and right cosets co-
incide; that is, if gH = Hg for all g ∈ G. We write
H ◁ G to mean that H is a normal subgroup of G.
Theorem 4.13. Let H be a subgroup of G. Then,
the following are equivalent conditions for H to be a
normal subgroup of G.
1. ghg −1 ∈ H ∀g ∈ G and h ∈ H.
2. gHg −1 = H ∀g ∈ G.
5 Homomorphism Corollary 5.9. If ϕ : G → H is a group homomor-
phism, then Ker(ϕ) is a normal subgroup of G.
Definition 5.1. Let (G, ⊗), (H, ⊕) be groups and
let ϕ : G → H be a function. Then ϕ is called a Theorem 5.10. A group homomorphism ϕ : G → H
homomorphism if ∀a, b ∈ G, we have is a 1 − 1 map if and only if Ker(ϕ) = {e}.
Theorem 5.11. Let ϕ : G → K be a group homomor-
ϕ(a ⊗ b) = ϕ(a) ⊕ ϕ(b) phism with kernel H. Then the left cosets of H form
a factor group, G/H where (aH)(bH) = (ab)H. Also
Definition 5.2. Let ϕ : G → H be a homomorphism, the map µ : G/H → ϕ[G] defined by µ(aH) = ϕ(a) is
where G, H are groups. Then ϕ is called an isomor- an isomorphism.
phism if it a one-to-one and onto functions. G and H
are said to be isomorphic, denoted by G ∼=H Theorem 5.12. Let H be a normal subgroup of G.
Then γ : G → G/H given by γ(x) = xH is a homo-
Definition 5.3. A 1-1 homomorphism is called a morphism with kernel H.
monomorphism and an onto homomorphism is
called an epimorphism. A homomorphism from a Theorem 5.13 (The Fundamental Homomorphism
group G to itself is called an endomorphism. An Theorem). Let ϕ : G → K be a group homomor-
isomorphism from a group G onto itself is called an phism with kernel H. Then ϕ[G] is a group and
automorphism. µ : G/H → ϕ[G] given by µ(gH) = ϕ(g) is an iso-
morphism. If γ : G → G/H is the homomorphism
Theorem 5.4. Let G, H and K be groups and let ϕ : given by γ(g) = gH, then ϕ(g) = (µ ◦ γ)(g) ∀ g ∈ G.
G → H and φ : H → K be functions.
Definition 5.14 (Inner Automorphism). The auto-
1. If ϕ and φ are homomorphisms, then φ ◦ ϕ : G → morphism ig : G → H where ig (x) = gxg −1 for all
K is a homomorphism. x ∈ G is the inner automorphism of G by g. Perform-
2. If ϕ and φ are homomorphisms, then φ ◦ ϕ is also ing ig on x is called conjugation of x by g.
an isomorphism. Remark 5.15. We have that
3. If ϕ is an ismorphism, then ϕ−1 : H → G is an gH = Hg ⇔ gHg −1 = H ⇔ ig [H] = H.
isomorphism.
Hence the normal subgroup of a group G are those that
Definition 5.5. Let ϕ be a mapping of a set X into a are invariant under all inner automorphisms.
set Y and let A ⊆ X and B ⊆ Y . The image of A in
Y under ϕ is the set ϕ[A] = {ϕ(a)| a ∈ A}. The set Definition 5.16. Let H < G and let N (H) = {g ∈
ϕ[X] is the range of ϕ. The inverse image ϕ−1 [B] of G| gHg −1 = H}. Then N (H) is called the normalizer
B in X is {x ∈ X| ϕ(x) ∈ B}. of H.

Theorem 5.6. Let ϕ be a homomorphism of a group Definition 5.17 (Simple Group). A group is simple
G into a group H. if it is nontrivial and has no proper nontrivial normal
subgroups.
1. If e is the identity element G, then ϕ(e) is the
Remark 5.18. Definition 5.17 says that a group G ̸=
′ ′
identity element e in H. That is, ϕ(e) = e .
{e} is simple if and only if the only normal subgroups
2. ∀a ∈ G and ∀n ∈ Z, ϕ(an ) = [ϕ(a)]n . In particu- of G are {e} and G.
lar, ϕ(a−1 ) = [ϕ(a)]−1 .
Definition 5.19 (Maximal Normal Subgroup). A
3. If H is a sungoup of G, then ϕ[H] is a subgroup maximal normal subgroup of a group G is a normal
′
of G . subgroup M ̸= G such that there is no proper subgroup
′ ′ ′
4. If K is a subgroup of G , then ϕ−1 [K ] is a sub- N of G properly containing M .
group of G. Theorem 5.20. M is a maximal normal subgroups of
Definition 5.7. Let ϕ : G → H be a homomorphism G if and oonly if G/M is simple.
′
of groups. The subgroup ϕ−1 [{e }] = {x ∈ G| ϕ(x) =
′
e } is the kernel of ϕ, denoted by Ker(ϕ).
6 Direct Products
Theorem 5.8. Let ϕ : G → H be a group of homo-
morphism and let H = Ker(ϕ). Let a ∈ G. Then the Definition 6.1. The cartesian produuct of sets
set ϕ−1 [{ϕ(a)}] = {x ∈ G| ϕ(x) = ϕ(a)} is the left S1 , S2 , . . . Sn is the set of all ordered n-tuples
coset aH of H and is also the right coset Ha of H. (a1 , a2 , . . . , an ), where ai ∈ Si for i = 1, 2, . . . , n. The
Cartesian product is denoted by S1 × S2 × · · · Sn or Definition 7.3. Let X be a G-set. If the identity ele-
Qn
i=1 Si . ment of G is the only element that leaves every x ∈ X
fixed, then G is said to act faithfully on X.
Theorem 6.2. Let G1 , G2 , . . . , Gn be groups. For
(a1 , a2 , . . . , an ) and (b1 , b2 , . . . , bn ) in ni=1 Gi , define Theorem 7.4. Let X be a G-set. Then Gx is a sub-
Q

(a1 , . . . , an )(b1 , . . . , bn ) to be element (a1 b1 , . . . , an bn ). group of G for each x ∈ X.

Then ni=1 Gi is a group under this binary operation. Definition 7.5 (Isotropy Subgroup). Let X be a G-
Q

Definition 6.3 (External Direct Product). Let set and let x ∈ X. The subgroup Gx is the isotropy
G1 , G2 , . . . , Gn be groups. For (stabilizer) subgroup of x in G.
Theorem 7.6. Let X be a G-set. For all x1 , x2 ∈ X,
(a1 , a2 , . . . , an ), (b1 , b2 , . . . , bn ) ∈ G1 × G2 × . . . Gn ,
let x1 ∼ x2 if and only if there exists g ∈ G such that
let gx1 = x2 . Then ∼ is an equivalence on X.
Definition 7.7 (Orbit). Let X be a G-set. Each cell
(a1 , . . . , an )(b1 , . . . , bn ) = (a1 b1 , . . . , an bn ).
in the partition of the equivalence relation in Theorem
The group G1 ×G2 ×. . . Gn is called the external direct ?? is an orbit in X under G. If x ∈ X, the cell con-
product of the groups G1 , G2 , . . . , Gn taining x is the orbit of x denoted by Gx. We obseve
that
Definition 6.4 (Internal Direct Product). A group G
is the internal direct product of its subgroups G1 ×G2 × Gx = {y ∈ X| y ∼ x}
. . . Gn if the function
= {y ∈ X| y = gx for some g ∈ G}
ϕ : G1 × G2 × . . . Gn → G = {gx| g ∈ G}
defined by letting
Theorem 7.8 (The Stabiliser-Orbit Theorem). Let X
ϕ(g1 , g2 , . . . , gn ) = g1 g2 · · · gn . be a G-set and let x ∈ X. Then |Gx| = (G : Gx ). If
|G| is finite, then |Gx| is a divisor of |G|.
is an isomorphism. We write G = G1 ⊠ G2 ⊠ · · · ⊠ Gn .
Definition 7.9 (The Class Equation).
Theorem 6.5. Let G = G1 × G2 × . . . Gn .
r
|G| = c +
X
1. If gi ∈ G for 1 ≤ i ≤ n and each gi has finite ni
order, then o((g1 , g2 , . . . , gn )) is the least common i=c+1

multiple of o(g1 ), o(g2 ), . . . o(gn ). is called the class equation of G. Each orbit in G under
2. If each G is a cyclic group of finite order, then G conjugation by G is called a conjugate class in G.
i
is cyclic if and only if |Gi | and |Gj | are relatively Theorem 7.10. Let p be a prime. Let G be a group
prime i ̸= j. of order pn with n ∈ Z+ , and let X be a finite G-set.
Then |X| ≡ |XG | (mod p)

7 Action of a Group on a Set

Definition 7.1 (Group Action on a Set). Let G be a
group and X a set. Then G is said to act on X if there
is a mapping ⋆ : G × X → X such that
1. e ⋆ x = x, for all x ∈ X.
2. (g1 g2 ) ⋆ x = g1 ⋆ (g2 ⋆ x), for all x ∈ X and for all
g1 , g2 ∈ G.
The mapping ⋆ is called an action of G on X and X
is called a G-set.
Theorem 7.2. Let X be a G-set. For each g ∈ G, the
function σg : X → X defined by σg (x) = gx for x ∈ X
is a permutation of X. The map ϕ : G → Sx defined
by ϕ(g) = σg is a homomorphism with the property
that ϕ(g)(x) = gx