Simple group

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

Examples

 * The easiest examples of simple groups are the simple abelian groups. An abelian group is simple if and only if it is cyclic of prime order.
 * The smallest non-abelian simple group is the alternating group on five letters. This is a group of order 60.
 * More generally, all alternating groups on five or more letters are simple. There are other infinite families of simple groups, primarily occurring as linear groups over fields.
 * The finite simple non-abelian groups come in 18 infinite families, and 26 exceptions, termed the sporadic simple groups.

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.

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 of a simple group is nothing distinguishing.

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

The testing problem
To determine on GAP whether a given group is simple:

IsSimpleGroup (group)

where group could either be a definition of a group or a name for a group already defined.

Textbook references

 * , Page 91
 * , Page 16
 * , Page 201, between points (2.3) and (2.4) (definition introduced in paragraph)