M417 Homework 7 Spring 2004
Instructions: Solutions are due Fri., April 2.
- 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.
- 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.
- 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.
- Show that Little Cayley is a corollary of Big Cayley.
- 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?