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Group Theory

Given G is a group, and H is a subgroup of G; for a,b in G, what does it mean for a to be congruent to b mod H?

Note by Siddharth Sabharwal
4 years, 2 months ago

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

Rohit Singh - 2 years ago

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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.

Mark Hennings - 4 years, 2 months ago

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