simple algebraic system
An algebraic system A is simple if the only congruences 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 subgroup
of G×G and an equivalence relation
on G at the same time. As an equivalence relation, C corresponds to a partition of G in the following manner: G=⋃i∈INi and C=⋃i∈IN2i, where Ni∩Nj=∅ for i≠j. Each of the Ni is an equivalence class
of C. Let N be the equivalence class containing 1. If a,b∈N, then [a]=[b]=[1], so that [ab]=[a][b]=[1][1]=[1], or ab∈N. In addition, [a-1]=[1][a-1]=[a][a-1]=[aa-1]=[1], so a1∈N. N is a subgroup of G. Furthermore, if c∈G, [cac-1]=[c][a][c-1]=[c][1][c-1]=[cc-1]=[1], so that cac-1∈N, N is a normal subgroup
of G. Conversely, given a normal subgroup N of G, forming left (right) cosets Ni of N, and taking C=⋃i∈IN2i 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 group. 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 |