Exercise · ZW-SUBGROUP-001 · ★☆☆ Basic
Which of the following sets is a subgroup of \((\mathbb{Z},+)\)?
Choose one option.
Show answer
- B. \(3\mathbb{Z}\), the multiples of 3
Explanation
The multiples of 3 contain 0, are closed under addition and each has its opposite in the set. The naturals lack opposites, the odd integers lack 0, and the last set is not closed under addition.
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-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-004 How many subgroups does (ℤ_12,+) have, counting the trivial one and the whole group?
- 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?