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6.15. Let G act on itself by conjugation as described in the proof of Theorem 6.25. Let x € G. The conjugacy class of x is the orbit of x for this action; i.e., it is the set {gxg-¹: ge G}. (a) Prove that G is the disjoint union of its conjugacy classes. (b) Prove that G is abelian if and only if each conjugacy classes of G contains a single element. (c) Describe the conjugacy classes of the dihedral group D3.

6.15. Let G act on itself by conjugation as described in the proof of Theorem 6.25. Let x € G. The conjugacy class of x is the orbit of x for this action; i.e., it is the set {gxg-¹: ge G}. (a) Prove that G is the disjoint union of its conjugacy classes. (b) Prove that G is abelian if and only if each conjugacy classes of G contains a single element. (c) Describe the conjugacy classes of the dihedral group D3.

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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6.15. Let G act on itself by conjugation as described in the proof of Theorem 6.25. Let x € G. The conjugacy class of x is the orbit of x for this action; i.e., it is the set {gxg-¹: g € G}. -1 (a) Prove that G is the disjoint union of its conjugacy classes. (b) Prove that G is abelian if and only if each conjugacy classes of G contains a single element. (c) Describe the conjugacy classes of the dihedral group D3. (d) Challenge Problem. Describe the conjugacy classes of the dihedral group Dn. (e) Describe the conjugacy classes of the quaternion group 2.
Transcribed Image Text:6.15. Let G act on itself by conjugation as described in the proof of Theorem 6.25. Let x € G. The conjugacy class of x is the orbit of x for this action; i.e., it is the set {gxg-¹: g € G}. -1 (a) Prove that G is the disjoint union of its conjugacy classes. (b) Prove that G is abelian if and only if each conjugacy classes of G contains a single element. (c) Describe the conjugacy classes of the dihedral group D3. (d) Challenge Problem. Describe the conjugacy classes of the dihedral group Dn. (e) Describe the conjugacy classes of the quaternion group 2.
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