Exercise · ZW-CYCLIC-GROUP-007 · ★★☆ Intermediate
Which of the following statements is true?
Choose one option.
Show answer
- D. Every cyclic group is abelian.
Explanation
Powers of \(g\) commute: \(g^ig^j=g^{i+j}=g^jg^i\). The others fail: the Klein group is abelian but not cyclic, \(\mathbb{Z}\) is cyclic and infinite, and \(\mathbb{Z}_5\) has four generators.
Review the concept: Cyclic group
A cyclic group is a group \(G\) with an element \(g\), called a generator, such that every element of \(G\) is a power of \(g\).
\[G=\langle g\rangle=\{\,g^k : k\in\mathbb{Z}\,\}\]
More practice problems: cyclic group
- ZW-CYCLIC-GROUP-001 How many generators does the cyclic group (ℤ_12,+) have?
- ZW-CYCLIC-GROUP-002 Which of the following elements generates (ℤ_10,+)?
- ZW-CYCLIC-GROUP-003 Which of the following groups is NOT cyclic?
- ZW-CYCLIC-GROUP-004 How many subgroups does (ℤ_18,+) have?
- ZW-CYCLIC-GROUP-005 Which of the following elements generates the group (ℤ_7^*,·)?
- ZW-CYCLIC-GROUP-006 How many elements does the subgroup of (ℤ_24,+) generated by 9 and 15 have?
- ZW-CYCLIC-GROUP-008 How many elements of order 4 does (ℤ_20,+) have?
- ZW-CYCLIC-GROUP-009 Let G=langle grangle be an infinite cyclic group. How many generators does G have?