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Talk:Conjugacy class

Reverting to previous text , as the centralizer is indeed a subgroup of G; if a,b in C_G(g), then:

C_G 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 C_G(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 C_G(g).

thus by elementary group theory, C_G(g) is a subgroup of G.

It is true that C_G(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

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