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**Theorem 7.1.1. [First Isomorphism Theorem]**
Let G be a group with normal subgroups N and H such that
N H.
Then H/N is a normal subgroup of G/N, and

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

**Theorem 7.1.2. [Second Isomorphism Theorem]**
Let G be a group, let N be a normal subgroup of G, and
let H be any subgroup of G.
Then HN is a subgroup of G,
HN
is a normal subgroup of H, and

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

**Theorem 7.1.3.**
Let G be a group with normal subgroups H, K such that HK = G and
HK = {e}. Then

G H **×** K .

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

**Definition 7.1.5.**
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 in G, where
i_{a} (x) = axa^{ -1}
for all x in 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).

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

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

is called the

**Proposition 7.1.8.**
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}^{×}

**15.**
Prove that a finite group whose only automorphism is the identity map
must have order at most two.
*Solution*

**16.**
Let p be a prime number,
and let A be a finite abelian group in which every element has order p.
Show that Aut(A) is isomorphic to
a group of matrices over **Z**_{p}.
*Solution*

**17.**
Let G be a group and let N be a normal subgroup of G of finite index.
Suppose that H is a finite subgroup of G
and that the order of H is relatively prime to the index of N in G.
Prove that H is contained in N.
*Solution*

**18.**
Let G be a finite group and let K be a normal subgroup of G
such that gcd (|K|,[G:K]) = 1.
Prove that K is a characteristic subgroup of G.
*Solution*

**19.**
A subgroup H of a finite group G is called a
**Hall subgroup** of G if its index in G
is relatively prime to its order.
That is, if gcd (|H|,[G:H]) = 1.
Prove that if H is a Hall subgroup of G
and N is any normal subgroup of G,
then (a) H
N is a Hall subgroup of N
and (b) HN / N is a Hall subgroup of G / N.
*Solution*

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