MHB Arjun's question at Yahoo Answers (Equivalence relation)

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The discussion addresses the equivalence relation defined on a group G with respect to a subgroup H, where two elements a and b are congruent modulo H if a * (b inverse) is in H. It establishes that this relation is reflexive, symmetric, and transitive, thus confirming it as an equivalence relation. The reflexive property is demonstrated by showing that any element a relates to itself since the identity element e is in H. The symmetric property is verified through the relationship between a and b, leading to the conclusion that if a is related to b, then b is related to a. Lastly, the transitive property is proven by combining the relationships among a, b, and c, confirming that if a is related to b and b to c, then a is also related to c.
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Here is the question:

Let G be a group and H be a subgroup of G. We say that a is congruent to b modulo H if and only if a* (b inverse) is an element of H. Show that congruence modulo H is an equivalence relation on G.

Here is a link to the question:

Equivalence Relation question? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 
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Hello Arjun,

Our relation is $a\sim b\Leftrightarrow ab^{-1}\in H$. This is an equivalence relation on $G$.

$(i)$ Reflexive. For all $a\in G$ is verified $aa^{-1}=e$, and $e\in H$ because $H$ is subgroup of $G$, so $a\sim a$.

$(ii)$ Symmetric. Using that $H$ is subgroup, $(xy)^{-1}=y^{-1}x^{-1}$ and $(x^{-1})^{-1}=x$:

$a\sim b\Rightarrow ab^{-1}\in H\Rightarrow (ab^{-1})^{-1}\in H\Rightarrow (b^{-1})^{-1}a^{-1}\in H\Rightarrow ba^{-1}\in H\Rightarrow b\sim a$

$(iii)$ Transitive. $a\sim b$ and $b\sim c$ implies $ab^{-1}\in H$ and $bc^{-1}\in H$. As $H$ is subgroup the product of these elements is in $H$, that is:
$$(ab^{-1})(bc^{-1})=a(b^{-1}b)c^{-1}=aec^{-1}=ac^{-1}\in H$$ which implies $a\sim c$. $\qquad \square$
 
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