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§ 7.1 Isomorphism Theorems; Automorphisms

 
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).

 
Proposition 7.1.6. 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).

 
Definition 7.1.7. For any group G, the subset

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

 
Proposition 7.1.8. For any group G, we have Inn(G) G/Z(G).

 
Example. 7.1.1. Aut(Z) Z₂ and Inn(Z) = {e}

 
Example. 7.1.2. Aut(Z_n) Z_n^×

§ 7.1 Isomorphism Theorems; Automorphisms: Solved problems

 
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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