# 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, 10 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

- 2 years, 8 months ago

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.

- 4 years, 10 months ago