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Prove that if B is a subgroup of G then the coset produced by multiplying every element of B with X s.t. X is a member of B is a group.

Prove that if B is a subgroup of G then the coset produced by multiplying every element of B with X s.t. X is a member of B is a group.

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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1- Prove that if B is a subgroup of G then the coset produced by multiplying every element of B with X s.t. X is a member of B is a group. 2- Prove that the element of an Abelian group are self-conjugate. 3- Prove that the number of classes in an Abelian group is equal to the number of its elements. 4- Verify that P(3) has the following class designation (I) E; (II) A, B, C; (III) D, F. 5- Verify that {E, D, F} is a self-conjugate subgroup of P(3). 6- Prove that the subgroups of an Abelian group are self-conjugate. 7- Show that a self-conjugate subgroup is composed of entire classes only.
Transcribed Image Text:1- Prove that if B is a subgroup of G then the coset produced by multiplying every element of B with X s.t. X is a member of B is a group. 2- Prove that the element of an Abelian group are self-conjugate. 3- Prove that the number of classes in an Abelian group is equal to the number of its elements. 4- Verify that P(3) has the following class designation (I) E; (II) A, B, C; (III) D, F. 5- Verify that {E, D, F} is a self-conjugate subgroup of P(3). 6- Prove that the subgroups of an Abelian group are self-conjugate. 7- Show that a self-conjugate subgroup is composed of entire classes only.
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