You may work with any ONE person on this homework set. If you choose to do that, please give the name of the person on your paper and definitely do write up the work to hand in SEPARATELY. I.e Your write-up must not be copied.

1. If g is an element of the group G and g has the property that for some other element x of the group it is true that g • x = x • g = x, then g is the identity element of G. (This is a “stronger” version of a theorem proved in class. Which theorem? In what sense is this a stronger result.)

2. Let G a group. Suppose o(a) = 2 for every a in G. Then show that G is an abelian group.

3. Let G be a group and H ⊂ G, H ≠ ∅. Prove H is a subgroup of G if and only if a, b ∈ H implies ab−1 ∈ H. (1 step subgroup test)

4. Let G be a group and let a ∈ G. Let C_G(a) = {x ∈ G | ax = xa}. Prove: C_G(a) is a subgroup of G. This subgroup is called the centralizer of a in G.

5. Suppose G is a group which has only one element a ∈ G such that o(a) = 2. Prove that ax = xa, for all x ∈ G.

6. Show that the intersection of any collection of subgroups of a group G is a subgroup.

7. Show that the union of subgroups may fail to be a subgroup.

8. Let G be a group. In class we showed that the “center” of G is a subgroup. Give a second proof of this fact based on problems 4 and 6.