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1. Let H and K be a subgroups of a group G. Define the relation on G as follows: ab if and only if a hbk, for some h E H and ke K Prove that is an equivalence relation on G.

1. Let H and K be a subgroups of a group G. Define the relation on G as follows: ab if and only if a hbk, for some h E H and ke K Prove that is an equivalence relation on G.

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. Let H and K be a subgroups of a group G. Define the relation on G as follows: ab if and only if a = hbk, for some h H and kek Prove that is an equivalence relation on G. 2. Let R+ denotes the set of positive real numbers and let f: R+ →R+ be the bijection defined by f(x) = 3r, for r> 0. Let denote the ordinary real number multiplication and let be the binary operation on R+ such that f: (R¹,.) → (R+, ) is a group isomorphism. (a) If x, y € R+, find a formula for ay. What is the identity element of (R+,)? (b) For ER+, find a formula for the inverse of r under . 3. If a group G has finitely many subgroups, does this mean that G is finite group? Justify your answer.
Transcribed Image Text:1. Let H and K be a subgroups of a group G. Define the relation on G as follows: ab if and only if a = hbk, for some h H and kek Prove that is an equivalence relation on G. 2. Let R+ denotes the set of positive real numbers and let f: R+ →R+ be the bijection defined by f(x) = 3r, for r> 0. Let denote the ordinary real number multiplication and let be the binary operation on R+ such that f: (R¹,.) → (R+, ) is a group isomorphism. (a) If x, y € R+, find a formula for ay. What is the identity element of (R+,)? (b) For ER+, find a formula for the inverse of r under . 3. If a group G has finitely many subgroups, does this mean that G is finite group? Justify your answer.
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