Abstract Math Analysis

1. Let G be a group.

(a) Given a, b ∈ G, prove that there exists a unique g ∈ G such that ag = b.

(b) Prove that G is abelian if and only if a^2 * b^2 = (ab)^2 for all a, b ∈ G.

2. Let G be a group.

(a) If H and K are subgroups of G, prove that H ⋂ K is a subgroup of G.

3. Let G and H be groups and f : G → H a group homomorphism.

(a) If e ∈ G is the identity element, prove that f(e) is the identity element for Rng(f).

(b) Given g ∈ G, prove that (f(g))^(-1) = f(g^(-1)).

(c) Prove that if f is bijective, then f^(-1) is a group homomorphism.

4. A subset S of a ring (R, +, ·) is a subring if (S, +, ·) is a ring with the same multiplicative identity as R. Prove that if S ⊆ R is closed under subtraction and multiplication (i.e. for every a; b ∈ S, a - b ∈ S and ab ∈ S) and 1 ∈ S, then S is a subring of R.

Sample Solution