Simple group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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Definition

Symbol-free definition

A group is said to be simple if the following equivalent conditions hold:

• It has no proper nontrivial normal subgroup
• Any homomorphism from it is either trivial or injective

Definition with symbols

A group $G$ is termed simple if the following equivalent conditions hold:

• For any normal subgroup $H$ of $G$, $H$ is either trivial or the whole group.
• Given any homomorphism $\phi :G$ → $H$ $\phi$ is either injective (that is, its kernel is trivial) or trivial (that is, it maps everything to the identity element).

In terms of the simple group operator

The group property of being simple is obtained by applying the simple group operator to the subgroup property of normality.

Property theory

Direct product-closed

A direct product of simple groups need not be simple.