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:
- $\RR^l_H := \set {\tuple {x, y} \in G \times G: x^{-1} y \in H}$
This is called left congruence modulo $H$.
Also known as
When $\tuple {x, y} \in \RR^l_H$, 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
- Left Congruence Modulo Subgroup is Equivalence Relation
- Definition:Left Coset
- Definition:Left Coset Space
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Theorem $11.1$
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.9$: Subgroups
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 37$
- 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $0$: Some Conventions and some Basic Facts
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 42$. Another approach to cosets
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $5$: Cosets and Lagrange's Theorem: Proposition $5.3$