Exercise · ZW-CENTER-OF-A-GROUP-007 · ★★☆ Intermediate
Which of the following statements about \(Z(G)\) is true for every group \(G\)?
Choose one option.
Show answer
- B. \(Z(G)\) is a subgroup of \(G\) and is abelian.
Explanation
The identity lies in the center, and products and inverses of central elements are central; moreover they commute with each other. The others fail: \(S_3\) has trivial center and every abelian group has \(Z(G)=G\).
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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- ZW-CENTER-OF-A-GROUP-008 Let G=ℤ_3× S_3, of order 18, with componentwise product. How many elements does Z(G) have?
- ZW-CENTER-OF-A-GROUP-009 Let x∈ Z(G) with x^2=e. For any g∈ G, what is (gx)^2?