Talk:Conjugacy class
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Reverting to previous text , as the centralizer is indeed a subgroup of G; if a,b in CG(g), then:
- CG is not empty, since (g*g)*g = g*(g*g).
- b * g = g * b → b -1 * g = g * b -1; thus b -1 is in CG(g)
- (a*b -1)*g = a*(b -1*g) = a*(g*b -1) = (a*g)*b -1 = (g*a)*b -1 = g*(a*b -1) → a*b -1 in CG(g).
thus by elementary group theory, CG(g) is a subgroup of G.
It is true that CG(g) is not a normal subgroup of G; but that's not what was stated in the article. Maybe this should be added to the article.User:chas_zzz_brown