Exercise · ZW-CENTER-OF-A-GROUP-001 · ★☆☆ Basic
How many elements does the center of the symmetric group \(S_3\) have?
Choose one option.
Show answer
- B. 1
Explanation
No permutation other than the identity commutes with all the others: a transposition does not commute with a 3-cycle, nor do two different transpositions. The center is \(\{e\}\).
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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