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|.