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In which of the following cases is H a subgroup of G? (a) G GLn (C) and H = GLn(R). (b) G = RX and H = {1, -1}. (c) G = Z+ and H is the set of positive integers. (d) G = RX and H is the set of positive reals. (e) G = GL₂(R) and H is the set of matrices a 0 [88]. 0 with a 0.

In which of the following cases is H a subgroup of G? (a) G GLn (C) and H = GLn(R). (b) G = RX and H = {1, -1}. (c) G = Z+ and H is the set of positive integers. (d) G = RX and H is the set of positive reals. (e) G = GL₂(R) and H is the set of matrices a 0 [88]. 0 with a 0.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Abstract Algebra

**2.4. In which of the following cases is \( H \) a subgroup of \( G \)?** (a) \( G = GL_n(\mathbb{C}) \) and \( H = GL_n(\mathbb{R}) \). (b) \( G = \mathbb{R}^\times \) and \( H = \{1, -1\} \). (c) \( G = \mathbb{Z}^+ \) and \( H \) is the set of positive integers. (d) \( G = \mathbb{R}^\times \) and \( H \) is the set of positive reals. (e) \( G = GL_2(\mathbb{R}) \) and \( H \) is the set of matrices \(\begin{bmatrix} a & 0 \\ 0 & 0 \end{bmatrix}\), with \( a \neq 0 \).
Transcribed Image Text:**2.4. In which of the following cases is \( H \) a subgroup of \( G \)?** (a) \( G = GL_n(\mathbb{C}) \) and \( H = GL_n(\mathbb{R}) \). (b) \( G = \mathbb{R}^\times \) and \( H = \{1, -1\} \). (c) \( G = \mathbb{Z}^+ \) and \( H \) is the set of positive integers. (d) \( G = \mathbb{R}^\times \) and \( H \) is the set of positive reals. (e) \( G = GL_2(\mathbb{R}) \) and \( H \) is the set of matrices \(\begin{bmatrix} a & 0 \\ 0 & 0 \end{bmatrix}\), with \( a \neq 0 \).
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