Exercise · ZW-SUBGROUP-008 · ★★☆ Intermediate
Which of these subsets of \(GL_2(\mathbb{R})\) is a subgroup?
Choose one option.
Show answer
- B. the matrices with determinant \(1\)
Explanation
The determinant is multiplicative: if \(\det A=\det B=1\) then \(\det(AB)=1\) and \(\det(A^{-1})=1\), and the identity qualifies. With determinant \(2\) or \(-1\) a product changes the determinant (4 or 1), and with \(|\det|\le1\) an inverse may have larger determinant.
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-007 The smallest subgroup of (ℤ,+) containing 4 and 6 is
- ZW-SUBGROUP-009 Which is a subgroup of (ℤ_7^*,·), the integers 1,dots,6 under multiplication mod 7?