Exercise · ZW-CENTER-OF-A-GROUP-010 · ★★★ Advanced
What is the center of the center, \(Z(Z(G))\)?
Choose one option.
Show answer
- D. \(Z(G)\)
Explanation
\(Z(G)\) is an abelian group, and in an abelian group the center is the whole group. Hence \(Z(Z(G))=Z(G)\).
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-007 Which of the following statements about Z(G) is true for every group G?
- 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?