# Homework Help: Normal Subgroups Proof with defined order

1. Apr 18, 2012

### tiger4

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. 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.

Last edited: Apr 18, 2012