- 7.1 Isomorphism theorems; automorphisms
- 7.2 Conjugacy
- 7.3 Groups acting on sets
- 7.4 The Sylow theorems
- 7.5 Finite abelian groups
- 7.6 Solvable groups
- 7.7 Simple groups

- 7.8 Nilpotent groups
- 7.9 Semidirect products
- 7.10 Classification of groups of small order

- Some group multiplication tables

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(G / N) / (H / N) G / H.

(HN) / N H / (H N).

G H **×** K .

**7.1.4. Proposition.**
Let G be a group and let
a G.
The function
i_{a} : G -> G
defined by
i_{a}(x) = axa^{-1}
for all x G is an isomorphism.

**7.1.5. Definition.**
Let G be a group. An isomorphism from G onto G is called an
** automorphism** of G.
An automorphism of G of the form
i_{a}, for some
a G, where
i_{a} (x) = axa^{-1}
for all
x G, is called an
** inner automorphism**
of G.
The set of all automorphisms of G will be denoted by Aut(G) and
the set of all inner automorphisms of G will be denoted by Inn(G).

**7.1.6. Proposition.**
Let G be a group.
Then Aut(G) is a group under composition of functions,
and Inn(G) is a normal subgroup of Aut(G).

**7.1.7. Definition.**
For any group G, the subset

Z(G) = { x G | xg = gx for all g G }

is called the
**7.1.8. Proposition.**
For any group G, we have Inn(G)
G/Z(G).

**Example.** 7.1.1.
Aut(**Z**) **Z**_{2}
and Inn(**Z**) = {e}

**Example.** 7.1.2.
Aut(**Z**_{n})
**Z**_{n}^{×}

If H and K are subgroups of G, then K is said to be a

**7.2.2. Proposition.**

**(a)**Conjugacy of elements defines an equivalence relation on any group G.**(b)**Conjugacy of subgroups defines an equivalence relation on the set of all subgroups of G.

**7.2.3. Definition.**
Let G be a group. For any element
x G, the set

{ a G | axa^{-1} = x }

For any subgroup H of G, the set

{ a G | aHa^{-1} = H }

**7.2.4. Proposition.**
Let G be a group and let
x G.
Then C(x) is a subgroup of G.

**7.2.5. Proposition.**
Let x be an element of the group G.
Then the elements of the conjugacy class of x
are in one-to-one correspondence
with the left cosets of the centralizer C(x) of x in G.

**Example.** 7.2.3.
Two permutations are conjugate in
S_{n}
if and only if they have the same shape
(i.e., the same number of disjoint cycles, of the same lengths).
Thus, in particular, cycles of the same length are always conjugate.

**7.2.6. Theorem. [Conjugacy class Equation]**
Let G be a finite group. Then

| G | = | Z(G) | + [ G : C(x) ]

where the sum ranges over one element x from each nontrivial conjugacy class.
**7.2.7. Definition.**
A group of order p^{n},
with p a prime number and
n 1,
is called a ** p-group.**

**7.2.8. Theorem. [Burnside]**
Let p be a prime number. The center of any p-group is nontrivial.

**7.2.9. Corollary.**
Any group of order p^{2} (where p is prime) is abelian.

**7.2.10. Theorem. [Cauchy]**
If G is a finite group and p is a prime divisor of the order of G,
then G contains an element of order p.

**(i)**a(bx) = (ab)x for all a,b G, and**(ii)**ex = x for the identity element e of G.

**7.3.2. Proposition.**
Let G be a group and let S be a set.
Any group homomorphism from G into the group Sym(S) of
all permutations of S defines an action of G on S.
Conversely, every action of G on S arises in this way.

**7.3.3. Definition.**
Let G be a group acting on the set S.
For each element x S, the set

Gx = { s S | s=ax for some a G }

is called the
G_{x}
= { a
G | ax = x }

S^{G}
= { x
S | ax = x for all a
G }

**7.3.4. Proposition.**
Let G be a group that acts on the set S, and let
xS.

**(a)**The stabilizer G_{x}of x in G is a subgroup of G.**(b)**There is a one-to-one correspondence between the elements of the orbit Gx of x under G and the left cosets of G_{x}in G.

**7.3.5. Proposition.**
Let G be a finite group acting on the set S.

**(a)**The orbits of S (under the action of G) partition S.**(b)**For any x S, | Gx | = [ G : G_{x}].

**7.3.6. Theorem.**
Let G be a finite group acting on the finite set S. Then

| S | = | S^{G} | +
_{}
[ G : G_{x} ],

**7.3.7. Lemma.**
Let G be a finite p-group acting on the finite set S. Then

| S |
| S^{G} | (mod p).

**7.3.8. Theorem. [Cauchy]**
If G is a finite group and p is a prime divisor of |G|, then
the number of solutions in G of the equation
x^{p} = e is a multiple of p.
In particular, G has an element of order p.

**7.4.2. Definition.**
Let G be a finite group, and let p be a prime number.
A subgroup P of G is called a
** Sylow p-subgroup**
of G if |P| = p^{k} for some integer
k 1
such that p^{k} is a divisor of |G| but p^{k+1} is not.

**7.4.3. Lemma.**
Let G be a finite group with
|G| = mp^{k},
where
k 1
and m is not divisible by p.
If P is a normal Sylow p-subgroup, then P contains every p-subgroup of G.

**7.4.4. Theorem. [Second and Third Sylow Theorems]**
Let G be a finite group of order n, and let p be a prime number.

**(a)**All Sylow p-subgroups of G are conjugate, and any p-subgroup of G is contained in a Sylow p-subgroup.**(b)**Let n = mp^{k}, with gcd(m,p)=1, and let s be the number of Sylow p-subgroups of G.

Then s | m and s 1 (mod p).

**7.4.5. Proposition.**
Let p > 2 be a prime, and let G be a group of order 2p.
Then G is either cyclic or isomorphic to the dihedral group
D_{p} of order 2p.

**7.4.6. Proposition.**
Let G be a group of order pq, where p > q are primes.

**(a)**If q is not a divisor of p-1, then G is cyclic.**(b)**If q is a divisor of p-1, then either G is cyclic or else G is generated by two elements a and b satisfying the following equations:

a^{p}= e, b^{q}= e, ba = a^{n}b

where n 1 (mod p) but n^{q}1 (mod p).

**7.5.2. Lemma.**
Let G be a finite abelian p-group, let
a G be an element of maximal order,
and let b<a> be any coset of G/<a>.
Then there exists
d G such that
d<a> = b<a> and
<a> <d> = {e}.

**7.5.3. Lemma.**
Let G be a finite abelian p-group.
If <a> is a cyclic subgroup of G of maximal order,
then there exists a subgroup H with
G <a> × H.

**7.5.4. Theorem. [Fundamental Theorem of Finite Abelian Groups]**
Any finite abelian group is isomorphic to a direct product
of cyclic groups of prime power order.
Any two such decompositions have the same number of factors of each order.

**7.5.5. Proposition.**
Let G be a finite abelian group.
Then G is isomorphic to a direct product of cyclic groups

**Z**_{n1}
×
**Z**_{n2}
× ** · · · ** ×
**Z**_{nk}

**7.5.6. Corollary.**
Let G be a finite abelian group.
If a G
is an element of maximal order in G,
then the order of every element of G is a divisor of the order of a.

**7.5.8 Theorem.**
Let p be an odd prime,
let k be a positive integer,
and let n = p^{k}. Then
**Z**_{n}^{×}
is a cyclic group.

**7.5.10. Theorem.**
If k 3,
and n = 2^{k},
then
**Z**_{n}^{×}
is isomorphic to the direct product
of a cyclic group of order 2 and a cyclic group of order
2^{k-2}.

**7.5.11. Corollary.**
The group
**Z**_{n}^{×}
is cyclic if and only if
n is of the form 2, 4,
p^{k}, or
2p^{k} for an odd prime p.

In elementary number theory,
an integer g is called a
** primitive root**
for the modulus n if
**Z**_{n}^{×}
is a cyclic group and [g]_{n}
is a generator for
**Z**_{n}^{×}.
Corollary 7.5.11 determines which moduli n have primitive roots.
The proof of Theorem 7.5.8
shows how to find a generator for
**Z**_{n}^{×},
where n = p^{k}.

G = N

**(i)**N_{i}is a normal subgroup in N_{i-1}for i = 1,2, . . . ,n,**(ii)**N_{i-1}/ N_{i}is abelian for i = 1,2, . . . ,n, and**(iii)**N_{n}= {e}.

**7.6.2. Proposition.**
A finite group G is solvable if and only if
there exists a finite chain of subgroups

G = N_{0}
N_{1}
** · · · **
N_{n} such that

**(i)**N_{i}is a normal subgroup in N_{i-1}for i = 1,2, . . . ,n,**(ii)**N_{i-1}/ N_{i}is cyclic of prime order for i = 1,2, . . . ,n, and**(iii)**N_{n}= {e}.

**7.6.3. Theorem.**
Let p be a prime number.
Any finite p-group is solvable.

**7.6.4. Definition.**
Let G be a group.
An element g G is called a
** commutator** if

g = aba^{-1}b^{-1}

for elements a,b G.
The smallest subgroup that contains all commutators of G is called the
** commutator subgroup**
or ** derived subgroup**
of G, and is denoted by G'.

**7.6.5. Proposition.**
Let G be a group with commutator subgroup G'.

**(a)**The subgroup G' is normal in G, and the factor group G/G' is abelian.**(b)**If N is any normal subgroup of G, then the factor group G/N is abelian if and only if G' N.

**7.6.6. Definition.**
Let G be a group.
The subgroup (G' )' is called the
** second derived subgroup**
of G. We define
G^{(k)}
inductively as
(G^{(k-1)})',
and call it the *k*
**th derived subgroup**

**7.6.7. Theorem.**
A group G is solvable if and only if
G^{(n)} = {e} for some positive integer n.

**7.6.8. Corollary.**
Let G be a group.

**(a)**If G is solvable, then so is any subgroup or homomorphic image of G.**(b)**If N is a normal subgroup of G such that both N and G/N are solvable, then G is solvable.

**7.6.9. Definition.**
Let G be a group.
A chain of subgroups

G =
N_{0}
N_{1}
. . .
N_{n} such that

**(i)**N_{i}is a normal subgroup in N_{i-1}for i = 1,2, . . . ,n,**(ii)**N_{i-1}/ N_{i}is simple for i = 1,2, . . . ,n, and**(iii)**N_{n}= {e}

**7.6.10. Theorem. [Jordan-Holder]**
Any two composition series for a finite group have the same length.
Furthermore, there exists a one-to-one correspondence between
composition factors of the two composition series under which
corresponding composition factors are isomorphic.

**7.7.2. Theorem.**
The symmetric group
S_{n} is not solvable for
n 5.

**7.7.3. Lemma.**
If n 4,
then no proper normal subgroup of
A_{n} contains a 3-cycle.

**7.7.4. Theorem.**
The alternating group A_{n} is simple if
n 5.

**7.7.5. Definition.**
Let F be a field.
The set of all
n × n
matrices with entries in F and determinant 1 is called the
** special linear group**
over F, and is denoted by
SL_{n}(F).

The group
SL_{n}(F)
modulo its center is called the
** projective special linear group**
and is denoted by
PSL_{n}(F).

**7.7.6. Proposition.**
For any field F, the center of
SL_{n}(F)
is the set of nonzero scalar matrices with determinant 1.

**Example.** 7.7.1.
PSL_{2}(F) S_{3}
if |F| = 2.

**Example.** 7.7.2.
PSL_{2}(F)
A_{4} if |F| = 3.

**7.7.7. Lemma.**
Let F be any field. Then
SL_{2}(F)
is generated by elements of the form

and
.

**7.7.8. Lemma.**
Let F be any finite field,
and let N be a normal subgroup of
SL_{2}(F).
If N contains an element of the form
with a 0,
then N = SL_{2}(F).

**7.7.9. Theorem.**
Let F be any finite field with |F| > 3.
Then the projective special linear group
PSL_{2}(F)
is a simple group.