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April 18th, 2012, 07:04 AM  #1 
Newbie Joined: Apr 2012 Posts: 16 Thanks: 0  Abstract Algebra normal groups. Quick please.
Let G be a group and H normal with G. Prove that if H = 2, then H is a subgroup of Z(G).

April 18th, 2012, 08:54 AM  #2 
Newbie Joined: Apr 2012 Posts: 16 Thanks: 0  Re: Abstract Algebra normal groups. Quick please.
Here's what i have come up with. Is this right? Since H is a subgroup, it must contain the identity, call it e. Call the nonidentity 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). 
April 18th, 2012, 01:13 PM  #3 
Senior Member Joined: Mar 2012 Posts: 294 Thanks: 88  Re: Abstract Algebra normal groups. Quick please.
your argument is correct. here is another: consider ghg^1 for any element g in G, where h is the nonidentity element of H. since H is normal, ghg^1 is in H. thus, either: ghg^1 = e > gh = g > h = e, contradiction! thus, this never happens. or: ghg^1 = h > gh = hg, for ALL g in G, hence h is in Z(G). since e is always in any subgroup, {e,h} = H is contained in Z(G). 

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