Re: Abstract
- From: magidin@xxxxxxxxxxxxxxxxx (Arturo Magidin)
- Date: Mon, 21 Nov 2005 01:51:15 +0000 (UTC)
In article <22655446.1132528760205.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
Eric <starwar636@xxxxxxx> wrote:
>A couple more hw problems I was hoping people could
>give me hints on.
>1. Order of G is pq where p and g are primes (not nexcessarily
>distinct). Prove that the order of the
>center of G is either 1 or pq.
Prove that if N is a subgroup of Z(G), and G/N is central, then G is
abelian (and therefore Z(G)=G).
The center of G must have order dividing pq. It is either 1, p, q, or
pq. If it were equal to p, what would be the order of G/Z(G)? And what
kind of group would it have to be? Reach a contradiction.
A similar argument will do if Z(G) is of order q.
>2. If H is a normal subgroup of G and order of H is 2,
>prove H is contained in the center of G.
If H = {1,a}, and g in G, you know that gag^{-1} must be in H. What
can it be equal to?
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx
.
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