M417 Homework 7 Spring 2004
Instructions: Solutions are due Fri., April 2.

1. If f: G -> H is a homomorphism of groups, and if B is a normal subgroup of H, show that f ^-1(B) is a normal subgroup of G.
2. If f: G -> H is a surjective homomorphism of groups, and if A is a normal subgroup of G, show that f(A) is a normal subgroup of H.
3. Give an example showing that the previous problem requires the hypothesis that f be surjective. [Hint: G is always a normal subgroup of itself.]
   
   Theorem 6.1 on p. 122 is sometimes called Little Cayley. Here is Big Cayley: Let A be a subgroup of a group G, and let P be the group of bijections from the set G/A of left cosets of A in G, to itself (i.e., P is the group of permutations of G/A). Then there is a homomorphism f: G -> P whose kernel is contained in A.
   
   
4. Show that Little Cayley is a corollary of Big Cayley.
5. Let A be a proper subgroup of a finite group G, such that p = |G : A| is the smallest prime dividing dividing |G|.
   • Use Big Cayley to prove that A is normal. [Hint: show that A = ker(f) by showing that |A : ker(f)| divides |P|, keeping in mind what prime factors |A : ker(f)| and |P| can have.]
   • Now show that any subgroup of index 2 in a finite group G is normal. Is this still true even if G is infinite?