Two elements \(x,y \in G\) are left congruent modulo the subgroup \(H\) if \(x^{-1}y \in H\), namely if \(xH = yH\), so that \(x\) and \(y\) define the same left coset. Left congruence is an equivalence relation on \(G\) - the decomposition of \(G\) into a disjoint union of equivalence classes (left cosets) gives us Lagrange's Theorem (when \(G\) is finite).

Two elements \(x,y \in G\) are right congruent modulo \(H\) if \(xy^{-1} \in H\), namely if \(Hx = Hy\), so that \(x\) and \(y\) define the same right coset. Right congruence is another equivalence relation on \(G\), which is different to left congruence unless the subgroup is normal.

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TopNewestThey belongs to the same class means b is conjugate to a iff at least one x exist in G such that b=xax–¹

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Two elements \(x,y \in G\) are

leftcongruent modulo the subgroup \(H\) if \(x^{-1}y \in H\), namely if \(xH = yH\), so that \(x\) and \(y\) define the same left coset. Left congruence is an equivalence relation on \(G\) - the decomposition of \(G\) into a disjoint union of equivalence classes (left cosets) gives us Lagrange's Theorem (when \(G\) is finite).Two elements \(x,y \in G\) are

rightcongruent modulo \(H\) if \(xy^{-1} \in H\), namely if \(Hx = Hy\), so that \(x\) and \(y\) define the same right coset. Right congruence is another equivalence relation on \(G\), which is different to left congruence unless the subgroup is normal.Log in to reply