Definition:Congruence Modulo Subgroup
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Definition
Let $G$ be a group.
Let $H$ be a subgroup of $G$.
We can use $H$ to define relations on $G$ as follows:
Left Congruence Modulo Subgroup
- $\RR^l_H := \set {\tuple {x, y} \in G \times G: x^{-1} y \in H}$
This is called left congruence modulo $H$.
Right Congruence Modulo Subgroup
- $\RR^r_H = \set {\tuple {x, y} \in G \times G: x y^{-1} \in H}$
This is called right congruence 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
- Results about congruence modulo a subgroup can be found here.
Sources
- 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): Appendix: Elementary set and number theory
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 20$. Cosets