Exercise · ZW-CENTER-OF-A-GROUP-003 · ★☆☆ Basic
The dihedral group \(D_4\) of symmetries of the square has order 8. How many elements does its center have?
Choose one option.
Show answer
- D. 2
Explanation
The center contains the identity and the \(180^\circ\) rotation, which commutes with every symmetry. Rotations by \(90^\circ\) and reflections do not commute with everything. The center has 2 elements.
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?
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- ZW-CENTER-OF-A-GROUP-009 Let x∈ Z(G) with x^2=e. For any g∈ G, what is (gx)^2?