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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