Normal Subgroups Proof with defined order

[tiger4]

Homework Statement

Let G be a group and H normal with G. Prove that if |H| = 2, then H is a subgroup of Z(G).

Homework Equations

The Attempt at a Solution

Since H is a subgroup, it must contain the identity, call it e. Call the non-identity element of H h. Thus, H = {h, e}. Since H contains its own inverses, h^2 = e (if h^2 = h, then h would have to be the identity).

Anyway, by the normality of H, we know that for any element g of G, gH = Hg. That is to say,
{gh, ge} = {hg, eg}, which means that
{gh, g} = {hg, g}

since h is not the identity, we know that gh≠g and hg≠g. Thus, in order for Hg and gH to be equal, we must have that gh = hg for an arbitrary element g. Thus, we have shown that H is a subgroup of Z(G).

a.) Is this right?
b.) If someone could clean up any loose ends, that would be great. My teacher is extremely picky on homework.

 

[mighty2000]

a.) Yes, your proof is correct.
b.) Here are some possible ways to clean up the proof:

- Instead of saying "Since H is a subgroup, it must contain the identity, call it e," you could say "Since H is a subgroup, it must contain the identity element, denoted by e."
- In the line "Thus, in order for Hg and gH to be equal, we must have that gh = hg for an arbitrary element g," you could add a bit more explanation. For example, you could say "Since H is a subgroup of G and H is normal, we know that for any element g of G, gH = Hg. This means that for any element h in H, gh must be equal to hg."
- In the line "since h is not the identity, we know that gh≠g and hg≠g," you could add a reason for why this is true. For example, you could say "Since |H| = 2, h is not equal to the identity element e. Therefore, gh cannot be equal to g, and hg cannot be equal to g."