Simple group

In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgroup and the corresponding quotient group. This process can be repeated, and for finite groups one eventually arrives at uniquely determined simple groups, by the Jordan–Hölder theorem.

The complete classification of finite simple groups, completed in 2004, is a major milestone in the history of mathematics.

Examples

Finite simple groups

The cyclic group

G

=

(

Z

/

3

Z

,

+

)

=

Z

3

{\displaystyle G=(\mathbb {Z} /3\mathbb {Z} ,+)=\mathbb {Z} _{3}}

of congruence classes modulo 3 (see modular arithmetic) is simple. If

H

{\displaystyle H}

is a subgroup of this group, its order (the number of elements) must be a divisor of the order of

G

{\displaystyle G}

which is 3. Since 3 is prime, its only divisors are 1 and 3, so either

H

{\displaystyle H}

is

G

{\displaystyle G}

, or

H

{\displaystyle H}

is the trivial group. On the other hand, the group

G

=

(

Z

/

12

Z

,

+

)

=

Z

12

{\displaystyle G=(\mathbb {Z} /12\mathbb {Z} ,+)=\mathbb {Z} _{12}}

is not simple. The set

H

{\displaystyle H}

of congruence classes of 0, 4, and 8 modulo 12 is a subgroup of order 3, and it is a normal subgroup since any subgroup of an abelian group is normal. Similarly, the additive group of the integers

(

Z

,

+

)

{\displaystyle (\mathbb {Z} ,+)}

is not simple; the set of even integers is a non-trivial proper normal subgroup.