Exercise · ZW-CENTER-OF-A-GROUP-005 · ★☆☆ Basic
A group \(G\) is abelian if and only if…
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- C. \(Z(G)=G\)
Explanation
\(G\) being abelian means every element commutes with every other, that is, every element lies in the center: \(Z(G)=G\).
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
- ZW-CENTER-OF-A-GROUP-001 How many elements does the center of the symmetric group S_3 have?
- ZW-CENTER-OF-A-GROUP-002 How many elements does the center of (ℤ_12,+) have?
- ZW-CENTER-OF-A-GROUP-003 The dihedral group D_4 of symmetries of the square has order 8. How many elements does its center h…
- ZW-CENTER-OF-A-GROUP-004 What is the center of the group GL_2(ℝ) of invertible 2×2 matrices?
- ZW-CENTER-OF-A-GROUP-006 The dihedral group D_6 of symmetries of the regular hexagon has order 12. How many elements does it…
- 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?
- ZW-CENTER-OF-A-GROUP-009 Let x∈ Z(G) with x^2=e. For any g∈ G, what is (gx)^2?