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 September 25th, 2007, 02:24 PM #1 Newbie   Joined: Sep 2007 Posts: 10 Thanks: 0 group centers Let G be a group. We say that an element b in G is in the center of G if bg=gb for every g in G. Let C be the set of all elements in the center. a) prove that C is a subgroup of G. b) prove that C is in fact a normal subgroup of G.
 September 28th, 2007, 05:43 PM #2 Global Moderator   Joined: Dec 2006 Posts: 20,099 Thanks: 1905 What progress have you made?
 October 1st, 2007, 07:36 AM #3 Newbie   Joined: Sep 2007 Posts: 10 Thanks: 0 i have finished part a, i am just stuck on part b, but here is what i have so far. The center of C of group G, is the set of all elements x in G for which xgx^-1=g for all g in G. This implies that gxg^-1=x for all g in G. So C is normal.
 October 1st, 2007, 05:19 PM #4 Global Moderator   Joined: Dec 2006 Posts: 20,099 Thanks: 1905 (b) Since every x in C satisfies xg = gx for very g in G, it follows (trivially) that gC = Cg for every g in G, so C is a normal subgroup.

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