Anstract Algebra proof - cosets

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Homework Help Overview

The discussion revolves around a proof in abstract algebra concerning the relationship between left and right cosets of a subgroup H in a group G. The original poster seeks to demonstrate that every left coset gH is equivalent to the right coset Hg under specific conditions regarding the subgroup.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the need to show a specific relationship between elements of the cosets, with one participant suggesting a rearrangement of terms to find a useful expression. There is also acknowledgment of confusion regarding the setup of the problem.

Discussion Status

The conversation indicates that some participants are beginning to clarify their understanding of the problem, with one expressing that a suggested approach makes sense. However, there is no explicit consensus or resolution yet.

Contextual Notes

Participants mention issues with notation and the clarity of the problem setup, indicating potential gaps in understanding that may affect their reasoning.

kathrynag
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Homework Statement


Let H be a subgroup of G such that g^-1hg is an element of H for all g in G and all h in H. Show that every left coset gH is the same as the right coset Hg.


Homework Equations





The Attempt at a Solution


need to show gh1=h2g
I know I need to show this, but am unsure on how to.
 
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I know the whole set up excpet for this part and it confuses me.
 
Let g be in G, h in H (sorry, Latex seems to be failing presently). As you say, you need to show that there exists k in H such that
gh = kg. You know that ghg-1 = k is in H. Rearrange this to get something useful.
 
Oh, that makes sense
 

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