Let x∈ Z(G) with x^2=e. For any g∈ G, what is (gx)^2?

Exercise · ZW-CENTER-OF-A-GROUP-009 · ★★★ Advanced

Let \(x\in Z(G)\) with \(x^2=e\). For any \(g\in G\), what is \((gx)^2\)?

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Since \(x\) commutes with \(g\), \((gx)^2=gxgx=g\,g\,xx=g^2x^2=g^2e=g^2\).

The center of a group \(G\) is the set \(Z(G)\) of elements of \(G\) that commute with every element of \(G\).

\[Z(G)=\{\,z\in G : zg=gz\ \text{for all } g\in G\,\}\]