# Simple group

A simple group is a mathematical object from the algebra. It is especially considered in group theory.

Each group has set itself and the neutral element as a normal subgroup. This raises the question of which groups have more or no further normal subgroups. The simple groups are by definition exactly those that hold only the normal subgroup mentioned two.

## Definition

A group is called simple if it has a normal subgroup only and with the neutral element. It is often also required.

## Finite simple groups

Finite simple groups are in group theory as the "basic building blocks" of finite groups, since every finite group can be constructed from simple groups in finitely many steps. Since 1982, the finite simple groups are completely classified, the list consists of

- Cyclic groups of prime order,
- Groups of Lie type, 16 each infinite series,
- Alternating groups with and
- 26 sporadic groups.

## Simple Lie Groups

Deviating from the usual in group theory above definition are referred to in the theory of Lie groups is a Lie group as easy if its Lie algebra is a simple Lie algebra. This is equivalent to the condition that all genuine normal subgroups are discrete subgroups. For example, SL (2, R ) is a simple group in the sense of Lie group theory, but has the normal subgroup. The quotient is a simple group in the sense of the usual definition in group theory.