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September 25th, 2007, 01:24 PM   #1
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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.
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September 28th, 2007, 04:43 PM   #2
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What progress have you made?
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October 1st, 2007, 06:36 AM   #3
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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.
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October 1st, 2007, 04:19 PM   #4
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(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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