Exercise · ZW-SUBGROUP-006 · ★★☆ Intermediate
Let \(H=2\mathbb{Z}\) and \(K=3\mathbb{Z}\) in \((\mathbb{Z},+)\). Which statement is true?
Choose one option.
Show answer
- C. \(H\cap K\) is a subgroup, but \(H\cup K\) is not
Explanation
The intersection \(H\cap K=6\mathbb{Z}\) is a subgroup (an intersection of subgroups always is). The union is not: \(2\) and \(3\) lie in it but \(2+3=5\) does not.
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-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?