simple algebraic system

An algebraic system A is simple if the only congruencesPlanetmathPlanetmathPlanetmathPlanetmath on it are A×A and Δ, the diagonal relation.

For example, let’s find out what are the simple algebras in the class of groups. Let G be a group that is simple in the sense defined above.

First, what are the congruences on G? A congruence C on G is a subgroupMathworldPlanetmathPlanetmath of G×G and an equivalence relationMathworldPlanetmath on G at the same time. As an equivalence relation, C corresponds to a partition of G in the following manner: G=iINi and C=iINi2, where NiNj= for ij. Each of the Ni is an equivalence classMathworldPlanetmath of C. Let N be the equivalence class containing 1. If a,bN, then [a]=[b]=[1], so that [ab]=[a][b]=[1][1]=[1], or abN. In addition, [a-1]=[1][a-1]=[a][a-1]=[aa-1]=[1], so a1N. N is a subgroup of G. Furthermore, if cG, [cac-1]=[c][a][c-1]=[c][1][c-1]=[cc-1]=[1], so that cac-1N, N is a normal subgroupMathworldPlanetmath of G. Conversely, given a normal subgroup N of G, forming left (right) cosets Ni of N, and taking C=iINi2 gives us the congruence C on G.

Now, if G is simple, then this says that the only congruences on G are G×G and Δ, which corresponds to G having G and 1 as the only normal subgroups. So, G as a simple algebra is just a simple groupMathworldPlanetmathPlanetmath. Conversely, if G is a simple group, then the only congruences on G are those corresponding to G and 1, the only normal subgroups of G. Therefore, a simple group is a simple algebra.

Remark. Any simple algebraic system is subdirectly irreducible.

Title simple algebraic system
Canonical name SimpleAlgebraicSystem
Date of creation 2013-03-22 16:46:56
Last modified on 2013-03-22 16:46:56
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 5
Author CWoo (3771)
Entry type Definition
Classification msc 08A30
Synonym simple
Defines simple algebra