# simple algebraic system

An algebraic system $A$ is *simple* if the only congruences^{} on it are $A\times A$ and $\mathrm{\Delta}$, 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 ${N}_{i}\cap {N}_{j}=\mathrm{\varnothing}$ for $i\ne j$. 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, $[{a}^{-1}]=[1][{a}^{-1}]=[a][{a}^{-1}]=[a{a}^{-1}]=[1]$, so ${a}^{1}\in N$. $N$ is a subgroup of $G$. Furthermore, if $c\in G$, $[ca{c}^{-1}]=[c][a][{c}^{-1}]=[c][1][{c}^{-1}]=[c{c}^{-1}]=[1]$, so that $ca{c}^{-1}\in N$, $N$ 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 $\mathrm{\Delta}$, which corresponds to $G$ having $G$ and $\u27e81\u27e9$ 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 $\u27e81\u27e9$, 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 |