Exercise · ZW-CENTER-OF-A-GROUP-008 · ★★☆ Intermediate

Let \(G=\mathbb{Z}_3\times S_3\), of order 18, with componentwise product. How many elements does \(Z(G)\) have?

Choose one option.

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  • B. 3

Explanation

A pair \((a,s)\) is central if each component is: \(a\) always is (\(\mathbb{Z}_3\) is abelian) and \(s\) must lie in \(Z(S_3)=\{e\}\). That leaves the 3 pairs \((a,e)\).

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Review the concept: Center of a group

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

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

More practice problems: center of a group

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