I need help on those questions please!

1. Let G be a group such that every element in G has order 2, i.e. for all g E G, 92 = 69. Prove that G is an abelian group. Recall, G is an abelian group if ab = but for all a, b E G. 2. Let G be a group, and x an element a E G. Dene the set G(a)={:cG:azz:=a:a} which is called the centralizer of a. (a) Consider the set of symmetries of the triangle D3. Determine the set 002240) (b) Let G be a. group and x an a, E G. Prove the set 0(a) is a subgroup of G. 3. Let H and K be subgroups of a group G. Prove that H ['1 K is also a subgroup of (3'. (Here, to show it is a subgroup is going to rely on the fact that H and K are each subgroups on their own). Recall, the intersection 2's H ['1 K = {g : g E H and g E K}