Subgroup of Abelian groups

[Heinsbergrelatz]

Ok im quite confused on wrapping my head around the concept of subgroups. I mean i understand the definition and the general requirements that is needed in order to be a subgroup, but im lost when it comes to doing exercises, just cant seem to do them. So can anyone help me solve these two problems im somehow stuck at?

 

• 1. if [latex]H=[ x \in G : x=y^2[/latex] for some [latex]y \in G ][/latex] prove that [latex]H[/latex] is a subgroup of [latex]G[/latex]
• 2. Let [latex]H[/latex] be a subgroup of [latex]G[/latex] and let [latex]K=[x\in G: x^{2} \in H][/latex], prove that [latex]K[/latex] is a subgroup of [latex]G[/latex] given that [latex]G[/latex] is Abelian. Why does [latex]K[/latex] fail to be a subgroup of [latex]G[/latex] if [latex]G[/latex] is not Abelian?
• 3. Prove that a Group [latex]G [/latex] is abelian if and only if for every [latex]a,b \in G[/latex] and positive integer n, [latex](ab)^n= a^n b^n[/latex]

Now i know these questions may seem fairly simple to many people but i would appreciate a clear explanation. Thank you

Edited June 30, 2012 by Heinsbergrelatz