A group can be complicated for two different reasons. It may have many elements, or it may resist quotient decompositions that preserve nonidentity information. The second kind of complexity is controlled by normal subgroups: whenever $N \trianglelefteq G$, the quotient $G/N$ records a piece of $G$ in which all elements of $N$ have been collapsed to the identity. If a nonidentity group has no [normal subgroup](/page/Normal%20Subgroup) available for this process, then quotient methods reach a hard endpoint. That endpoint is a simple group.

The word simple is slightly misleading at first encounter. A simple group need not be easy to understand; $A_5$ already contains enough structure to encode the rotational symmetries of the icosahedron. The point is not computational simplicity. The point is indivisibility with respect to normal subgroups. A simple group is a group with no nonidentity quotient decomposition coming from an internal normal subgroup.

This page develops the general definition, but most of its tests and examples are finite-group tests. Infinite simple groups exist and are important, especially in geometric group theory and Lie theory, but the first structural story is clearest in the finite setting, where orders, Sylow subgroups, and conjugacy-class counts turn normality into arithmetic.

[example: Why Normality Is the Right Test] Let $G=S_3$, and let $H=\{e,(12)\}$. The subgroup $H$ has order $2$, since it contains exactly the identity and the transposition $(12)$. To test normality, conjugate its nonidentity element by $(123)$. Since $(123)^{-1}=(132)$, we compute \begin{align*} (123)(12)(123)^{-1}=(123)(12)(132)=(23). \end{align*} The result $(23)$ is not in $H$, so $H$ is not stable under conjugation by elements of $S_3$. Thus $H$ is not normal in $S_3$. Equivalently, the left and right cosets by $(123)$ do not match: \begin{align*} (123)H=\{(123),(123)(12)\}=\{(123),(13)\}. \end{align*} \begin{align*} H(123)=\{(123),(12)(123)\}=\{(123),(23)\}. \end{align*} Because these cosets are different, $H$ cannot be used as the subgroup collapsed in a [quotient group](/theorems/790) $S_3/H$. By contrast, $A_3=\{e,(123),(132)\}$ is normal in $S_3$. Indeed, for any $\sigma\in S_3$, \begin{align*} \sigma(123)\sigma^{-1}=(\sigma(1)\ \sigma(2)\ \sigma(3)), \end{align*} which is either $(123)$ or $(132)$, and the same statement holds for the inverse cycle $(132)$. Hence conjugation by every element of $S_3$ sends elements of $A_3$ back into $A_3$, so $A_3\trianglelefteq S_3$. The quotient has exactly two cosets, \begin{align*} A_3=\{e,(123),(132)\} \end{align*} and \begin{align*} (12)A_3=\{(12),(23),(13)\}, \end{align*} so $|S_3/A_3|=2$. This is why arbitrary subgroups do not measure quotient decomposability: quotient groups come from normal subgroups, because normality is precisely the condition that makes coset multiplication well-defined. [/example]

The example also shows that the definition of simple group belongs under the broader definition-family of [group](/page/Group): it is not a new kind of algebraic operation, but a special condition on the normal subgroup structure of an existing group.

## Definition

### The Indivisible Case

The obstruction to decomposing a group by quotients is the absence of proper nonidentity normal subgroups. The identity subgroup is always normal, and the whole group is always normal, so these two unavoidable cases must be excluded from the test.

[definition: Simple Group] A group $G$ is simple if $G$ is nonidentity and the only normal subgroups of $G$ are $\{e\}$ and $G$. [/definition]

This definition is deliberately about normal subgroups rather than all subgroups. A group may have many subgroups and still be simple. What it cannot have is a subgroup that is stable under every conjugation action of $G$ and therefore capable of serving as the kernel of a quotient map.

The condition $G \ne \{e\}$ is included so that the identity group does not become an awkward degenerate simple group. In decomposition theory, simple groups are intended to be the indivisible non-zero pieces, much as prime numbers are the indivisible positive integers greater than $1$.

### Kernels and Quotients

The most effective way to use simplicity is not usually to list normal subgroups directly. Kernels of homomorphisms are automatically normal, so every attempted homomorphic image of a simple group faces a two-way test: either nothing except the identity is killed, or the whole source is killed.

[quotetheorem:10103]

The theorem says that a homomorphism out of a simple group has only two possibilities: it is injective, or it collapses all of $G$ to the identity element of the target. Thus a simple group cannot map non-faithfully onto a proper quotient while retaining nonidentity information.

The same idea can be phrased through quotient groups, but now the normal subgroup must be named explicitly: quotient groups of $G$ are precisely the groups $G/N$ with $N \trianglelefteq G$. Simplicity says that this construction has only the two unavoidable outcomes.

[quotetheorem:10104]

This makes simple groups the terminal objects for quotient-based decomposition. A non-simple group exposes a proper normal subgroup $N$, and the quotient $G/N$ is a nonidentity quotient image in which the elements of $N$ have been identified with the identity.

## Normal Subgroups as Quotient Data

### Minimal Normal Structure