Simple group

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This article is about a term related to the Classification of finite simple groups

Definition

Symbol-free definition

A nontrivial 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 nontrivial 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.

Relation with other properties

This property is a pivotal (important) member of its property space. Its variations, opposites, and other properties related to it and defined using it are often studied

• Category: Variations of simplicity
• Category: Opposites of simplicity

Stronger properties

• Absolutely simple group
• Strictly simple group
• Simple non-Abelian group
• Finite simple group
• Finite simple non-Abelian group

Weaker properties

• Characteristically simple group
• Directly indecomposable group
• Semidirectly indecomposable group
• Primitive group
• Monolithic group
• One-headed group

Facts

Proper subgroups are core-free

In a simple group, the normal core of any subgroup is a normal subgroup, and hence is either the whole group or the trivial subgroup. Thus, the normal core of any proper subgroup must be the trivial subgroup.

In other words, every proper subgroup is core-free.

Nontrivial subgroups are contranormal

In a simple group, the normal closure of any subgroup is either the whole group or the trivial subgroup. Thus, the normal closure of any nontrivial subgroup is the whole group.

In other words, every nontrivial subgroup of a simple group is contranormal.

Subgroup-defining functions collapse to trivial or improper subgroup

Any subgroup-defining function (such as the center, the commutator subgroup, the Frattini subgroup) returns a characteristic subgroup of the whole group. In other words, the center, commutator subgroup, Frattini subgroup etc. are all characteristic subgroups.

Since every characteristic subgroup is normal, each of these is also a normal subgroup. But when the whole group is simple, this forces each of these to be either the trivial subgroup or the whole group. Thus, for instance:

• The center of any simple group is either trivial or the whole group. Hence, every simple group is either centerless or Abelian.
• The commutator subgroup of any simple group is either trivial or the whole group. Hence, every simple group is either Abelian or perfect.

The only simple Abelian groups are cyclic groups of prime order

The proof of this follows more or less directly from the fact that in a simple Abelian group, every subgroup is normal, and hence, the subgroup generated by any nonidentity element is normal. This forces that the whole group is cyclic generated by any element, and hence it must be cyclic of prime order.

Metaproperties

Direct products

A direct product of simple groups is not simple. In fact, the two direct factors are themselves normal subgroups.

Subgroups

Every finite group occurs as a subgroup of some simple group. Hence the property of being embeddable as a subgroup o f a simple group is nothing distinguishing.

Quotients

The only quotients of a simple group are itself and the trivial group.