Definition:Congruence Modulo Subgroup/Left Congruence

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Definition

Let $G$ be a group.

Let $H$ be a subgroup of $G$.


We can use $H$ to define a relation on $G$ as follows:

$\mathcal R^l_H := \set {\tuple {x, y} \in G \times G: x^{-1} y \in H}$

This is called left congruence modulo $H$.


Also known as

When $\tuple {x, y} \in \mathcal R^l_H$, we write:

$x \equiv^l y \pmod H$

which is read: $x$ is left congruent to $y$ modulo $H$.


Additive Group of Ring

Some authors introduce the concept of congruence modulo $H$ in the context of ring theory.

In this case, the group $G$ is taken to be the additive group of a ring.

This is acceptable, but such a treatment does presuppose that $G$ is abelian.

In such a context, all the richness of the analysis of normal subgroups is disappointingly bypassed.


Also see


Sources