# simple algebraic system

An algebraic system $A$ is simple if the only congruences on it are $A\times A$ and $\Delta$, 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 $G\times G$ 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: $G=\bigcup_{i\in I}N_{i}$ and $C=\bigcup_{i\in I}N_{i}^{2}$, where $N_{i}\cap N_{j}=\varnothing$ for $i\neq j$. Each of the $N_{i}$ is an equivalence class of $C$. Let $N$ be the equivalence class containing $1$. If $a,b\in N$, then $[a]=[b]=[1]$, so that $[ab]=[a][b]=[1][1]=[1]$, or $ab\in N$. In addition, $[a^{-1}]=[1][a^{-1}]=[a][a^{-1}]=[aa^{-1}]=[1]$, so $a^{1}\in N$. $N$ is a subgroup of $G$. Furthermore, if $c\in G$, $[cac^{-1}]=[c][a][c^{-1}]=[c][1][c^{-1}]=[cc^{-1}]=[1]$, so that $cac^{-1}\in N$, $N$ is a normal subgroup of $G$. Conversely, given a normal subgroup $N$ of $G$, forming left (right) cosets $N_{i}$ of $N$, and taking $C=\bigcup_{i\in I}N_{i}^{2}$ gives us the congruence $C$ on $G$.

Now, if $G$ is simple, then this says that the only congruences on $G$ are $G\times G$ and $\Delta$, which corresponds to $G$ having $G$ and $\langle 1\rangle$ 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 $\langle 1\rangle$, 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 SimpleAlgebraicSystem 2013-03-22 16:46:56 2013-03-22 16:46:56 CWoo (3771) CWoo (3771) 5 CWoo (3771) Definition msc 08A30 simple simple algebra