Download 1. (1.6.17) Let G be any group. Prove that the map from G to itself

Math 332: Abstract Algebra
Mckenzie West
Theory Assignment 2
Due 2/10
1. (1.6.17) Let G be any group. Prove that the map from G to itself defined by g 7→ g −1
is a homomorphism if and only if G is abelian.
2. (1.7.18) Let H be a group acting on a set A. Prove that the relation ∼ on A defined by
a ∼ b if and only if a = h · b for some h ∈ H
is an equivalence relation.
For each x ∈ A, the equivalence class of x under ∼ is called the orbit of x under the
action of H.
3. (1.7.19) Let H be a subgroup of the finite group G and let H act on G by left multiplication. Let x ∈ G and let Ox be the orbit of x under the action of H. Prove that the
map
H → Ox defined by h 7→ hx
is a bijection. This implies that all orbits have cardinality H. From this and problem
(1.7.18) above, deduce the following theorem:
Theorem (Lagrange’s Theorem). If G is a finite group and H is a subgroup of G then
|H| divides |G|.
4. (2.1.8) Let H and K be subgroups of G. Prove that H ∪ K is a subgroup of G if and
only if H ⊆ K or K ⊆ H.
1