Centralizer is inclusion-reversing

Let G be a group. Prove that if A and B are subsets of G with A \subseteq B, then C_G(B) is a subgroup of C_G(A).


Let x \in C_G(B). Then for all b \in B, xbx^{-1} = b. Since A \subseteq B, for all a \in A we have xax^{-1} = a, so that x \in C_G(A). Thus C_G(B) \subseteq C_G(A), and hence C_G(B) \leq C_G(A).

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