Exercise · ZW-CYCLIC-GROUP-008 · ★★☆ Intermediate
How many elements of order 4 does \((\mathbb{Z}_{20},+)\) have?
Choose one option.
Show answer
- A. 2
Explanation
A cyclic group has a unique subgroup of order 4, and it has \(\varphi(4)=2\) generators, which are precisely the elements of order 4: \(5\) and \(15\).
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-007 Which of the following statements is true?
- ZW-CYCLIC-GROUP-009 Let G=langle grangle be an infinite cyclic group. How many generators does G have?