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Definition

Standard definition

A simple group is a group $G$ with exactly two quotient groups: the trivial quotient group $\{1\}\cong G/G$ and the group $G\cong G/\{1\}$ itself.

Equivalently, a simple group is a group possessing exactly two normal subgroups: the trivial subgroup $\{1\}$ and the group $G$ itself. One can also say that a normal subgroup is trivial iff it is not $G$ (compare the definition in constructive mathematics below).

Note that the trivial group does not itself count as simple, on the grounds that it has only one quotient group (or only one normal subgroup). It may be possible to find authors that use “at most” in place of “exactly”, thereby allowing the trivial group to be simple. (Compare too simple to be simple.)

In constructive algebra

In constructive mathematics, we consider a group $G$ equipped with a tight apartness $\ne$ such that the group operations are strongly extensional. Then $G$ is simple if, given any normal antisubgroup $A$ of $G$, $A$ owns every nonidentity element (every $x$ such that $x\ne 1$) iff $A$ is inhabited. In other words, $A$ is the $\ne$-complement of the identity subgroup $\{1\}$ iff $A$ is apart from the $\ne$-complement $\varnothing$ of the improper subgroup $G$ in the sense that the symmetric difference of $A$ and $\varnothing$ is inhabited. (Replacing ‘iff’ with ‘if’ here would allow the trival group to be simple.)

Examples

Finite simple groups

Simple groups are most commonly encountered in the theory of finite groups. Every finite group $G$ admits a composition series?, i.e., a finite filtration of subgroups

where each inclusion $G_i\subseteq G_{i+1}$ is a normal subgroup and the quotient $G_{i+1}/G_i$ (called a composition factor) is simple. The condition of simplicity means that that the filtration cannot be further refined by addition of strict inclusions of normal subgroups. Furthermore, the Jordan-Hölder theorem? ensures that any two composition series have the same length and the same composition factors (up to permutation).

Thus finite simple groups are in some sense the primitive building blocks of finite groups generally. The massive program of classifying all finite simple groups was announced as completed by Daniel Gorenstein in 1983, although some doubts remained because there were some gaps in proofs. Most if not all the gaps are considered by experts in the area to have been filled, but there remain some notable skeptics, including for example Jean-Pierre Serre and John H. Conway (verification needed here). See classification of finite simple groups.

Infinite simple groups

There are simple groups of any cardinality $\kappa$; take for example the smallest normal subgroup of the automorphism group $\mathrm{Aut}(\kappa )$ containing all 3-cycles (this is the infinite version of the alternating group?).