Abstract Algebra: Theory and Applications · Problem 14.5.22

The Sylow Theorems — Thomas Judson

Mathematics — Abstract Algebra · centralizer-conjugation

Problem

Let a∈Ga \in G. Show that for any g∈Gg \in G, gC(a)g−1=C(gag−1)gC(a)g^{-1} = C(gag^{-1}). Hint. Use the fact that x∈gC(a)g−1x \in gC(a)g^{-1} if and only if g−1xg∈C(a)g^{-1}xg \in C(a).

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