Let φ : G1 → G2 be a group homomorphism. (abstract algebra)
(a) Suppose H is a subgroup of G1. Define φ(H) = {φ(h) | h ∈ H}.
Prove that φ(H) is a subgroup of G2.
(b) Let ker(φ) = {g ∈ G1 | φ(g) = e2}. Prove that ker(φ) is a
subgroup of G1.
(c) Prove that φ is a group isomorphism if and only if ker(φ) =
{e1} and φ(G1) = G2.
Abstract Algebra
For the group S4, let H be the subset of all permutations that fix
the element 4.
a) show this is a subgroup
b) describe an isomorphism from S3 to H
Let (G,+) be an abelian group and U a subgroup of G. Prove that
G is the direct product of U and V (where V a subgroup of G) if
only if there is a homomorphism f : G → U with f|U =
IdU
Let G be a cyclic group generated by an element a.
a) Prove that if an = e for some n ∈ Z, then G is
finite.
b) Prove that if G is an infinite cyclic group then it contains
no nontrivial finite subgroups. (Hint: use part (a))
Let G be a group. For each x ∈ G and a,b ∈ Z+
a) prove that xa+b = xaxb
b) prove that (xa)-1 = x-a
c) establish part a) for arbitrary integers a and b in Z
(positive, negative or zero)