# Proof of union of subgroups as a subgroup

1. Jul 13, 2008

### needhelp83

Prove that if (H,o) and (K,o) are subgroups of a group (G,o), then (H $$\cap$$ K,o) is a subgroup of (G,o).

Proof:
The identity e of G is in H and K, so e $$\in$$ H$$\cap$$K and H$$\cap$$K is not empty. Assume j,k $$\in$$ H$$\cap$$K. Thus jk$$^{-1}$$ is in H and K, since j and k are in H and K. Therefore, jk$$^{-1}$$ $$\in$$ H $$\cap$$ K making H$$\cap$$K a subgroup.

Just trying to check this proof and see if I did a a good job at it.

Let (G,o) be an Abelian group and let a,b $$\in$$ G, Prove that (a o b)$$^{2}$$=a$$^{2}$$ o b$$^{2}$$

Not sure where to begin with this proof. Any help?

2. Jul 13, 2008

### matness

Re: Proofs

The first one is fine
For the second just use definition of ^2
(a*b)^2= (a*b)*(a*b)
and definition of abelian groups i.e cahnging places of elements does not effeect anything

3. Jul 14, 2008

### needhelp83

Re: Proofs

Thanks for the help matness