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efine xHx-1= {xhxh Hx is a subgroup of G. His cyclic, then xHx E H is cyc

efine xHx-1= {xhxh Hx is a subgroup of G. His cyclic, then xHx E H is cyc

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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1. Let G be a group and let H be a subgroup of G. For any fixed x in G, define xHx1 = {xhx-¹|hEH). Prove the following. a. xHx¹ is a subgroup of G. b. If His cyclic, then xHx-¹ is cyclic. c. If His Abelian, then xHx-¹ is Abelian. The group xHx¹ is called a conjugate of H. (Note that conjuga- tion preserves structure.)
Transcribed Image Text:1. Let G be a group and let H be a subgroup of G. For any fixed x in G, define xHx1 = {xhx-¹|hEH). Prove the following. a. xHx¹ is a subgroup of G. b. If His cyclic, then xHx-¹ is cyclic. c. If His Abelian, then xHx-¹ is Abelian. The group xHx¹ is called a conjugate of H. (Note that conjuga- tion preserves structure.)
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