Exercise · ZW-SUBGROUP-002 · ★☆☆ Basic
What condition is enough for a subset \(H\) of a group \(G\) to be a subgroup?
Choose one option.
Show answer
- C. \(H\) is nonempty and \(ab^{-1}\in H\) for all \(a,b\in H\)
Explanation
If \(H\) is nonempty and \(ab^{-1}\in H\), taking \(a=b\) gives the identity, then inverses, then products. Closure plus the identity alone can miss inverses, as in \(\mathbb{N}\subset\mathbb{Z}\).
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-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?