Exercise · ZW-P-GROUP-010 · ★★★ Advanced
Which cannot be the order of the center of a group of order \(27\)?
Choose one option.
Show answer
- D. \(1\)
Explanation
A nontrivial \(p\)-group has a nontrivial center: the class equation gives \(27=|Z|+\) (multiples of \(3\)), so \(3\) divides \(|Z|\). Hence the center cannot have a single element.
Review the concept: p-group
A \(p\)-group is a finite group \(G\) whose order is a power \(p^n\), with \(n\ge1\), of a prime number \(p\).
\[|G|=p^{n}\]
More practice problems: p-group
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