Exercise · ZW-SUBGROUP-004 · ★★☆ Intermediate
How many subgroups does \((\mathbb{Z}_{12},+)\) have, counting the trivial one and the whole group?
Choose one option.
Show answer
- B. 6
Explanation
The subgroups of \(\mathbb{Z}_{12}\) are those generated by a divisor of 12: \(\{0\}\), \(\langle 6\rangle\), \(\langle 4\rangle\), \(\langle 3\rangle\), \(\langle 2\rangle\) and the whole group; 12 has 6 divisors.
Review the concept: Subgroup
A subset \(H\) of a group \(G\) that is itself a group under the operation of \(G\).
\[H\le G\]
More practice problems: subgroup
- ZW-SUBGROUP-001 Which of the following sets is a subgroup of (ℤ,+)?
- ZW-SUBGROUP-002 What condition is enough for a subset H of a group G to be a subgroup?
- ZW-SUBGROUP-003 Which of the following is NOT a subgroup of (ℤ_12,+)?
- ZW-SUBGROUP-005 Which is a subgroup of (ℝ^*,·), the nonzero reals under multiplication?
- ZW-SUBGROUP-006 Let H=2ℤ and K=3ℤ in (ℤ,+). Which statement is true?
- ZW-SUBGROUP-007 The smallest subgroup of (ℤ,+) containing 4 and 6 is
- ZW-SUBGROUP-008 Which of these subsets of GL_2(ℝ) is a subgroup?
- ZW-SUBGROUP-009 Which is a subgroup of (ℤ_7^*,·), the integers 1,dots,6 under multiplication mod 7?