Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question

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 \( a \neq 0 \).
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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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