How to prove an equivalence relation without knowledge of the group?

Asked Jan 04 '18 at 17:21

Active Jan 04 '18 at 17:37

Viewed 29 times

"Let $x,y$ be elements of the group G. Prove the following is an equivalence relation: $x$ is related to $y$ iff there exists some $g$ in $G$ such that $y = g \cdot x \cdot g^{-1}$"

Now I understand that I must prove that the relation is reflexive,symmetric and transitive. But without knowledge of the set or whether the group binary operation is commutative I'm getting very confused on how to do it. I know that I just need to use the group properties of associativity, closure invertibility and identity, but I'm unsure how.

Any help appreciated.

edited Jan 04 '18 at 17:37

Alex Vong

1,654

asked Jan 04 '18 at 17:21

C. Hart

You don't need to know what the values are to work with the property. Write down what the definitions mean in terms of the relation. – Michael Burr Jan 04 '18 at 17:23
That was fast Dietrich Burde. – For the love of maths Jan 04 '18 at 17:23
For conjugate elements see here. – Dietrich Burde Jan 04 '18 at 17:24
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@MohammadZuhairKhan Some questions appear so often here that it takes no time to answer them:) – Dietrich Burde Jan 04 '18 at 17:27
Well you did not get that badge for nothing ;) – For the love of maths Jan 04 '18 at 17:29

How to prove an equivalence relation without knowledge of the group?

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