Exercise · ZW-SUBGROUP-007 · ★★☆ Intermediate
The smallest subgroup of \((\mathbb{Z},+)\) containing \(4\) and \(6\) is
Choose one option.
Show answer
- D. \(2\mathbb{Z}\)
Explanation
A subgroup containing 4 and 6 contains \(6-4=2\), and \(2\mathbb{Z}\) contains both. In general the smallest subgroup with \(a\) and \(b\) is \(\gcd(a,b)\mathbb{Z}\), not the least common multiple.
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-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-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?