Perfect group

In mathematics, perfect groups are those groups that are identical to their commutator group .

A group is therefore perfect if where where denotes the commutator group. In the past, perfect groups were also called perfect groups. ${\displaystyle G}$ ${\displaystyle G=[G,G]}$ ${\displaystyle [G,G]}$

properties

Note: The properties presented here refer to non- trivial perfect groups.

Factor groups of perfect groups are perfect. Since every commutative factor group factors out the commutator group, perfect groups have no non-trivial Abelian factor groups. So perfect groups are highly not Abelian, since the commutator group is the smallest normal divisor , so that the associated factor group is Abelian. In particular, perfect groups are therefore not resolvable and therefore do not have any resolvable factor groups.

Examples

The alternating groups are perfect for , because they are even simple , so they have no non-trivial normal divisors, and they are not Abelian, so the commutator group is the entire group. Since the commutator group is identical to the small group of four , is not perfect. The Abelian group is simple but not perfect because as an Abelian group it owns as a commutator group. ${\displaystyle A_{n}}$ ${\displaystyle n\geq 5}$ ${\displaystyle [A_{4},A_{4}]}$ ${\displaystyle V}$ ${\displaystyle A_{4}}$ ${\displaystyle A_{3}}$ ${\displaystyle \{1\}}$

If a non-Abelian group is easy, then it is perfect. Because the commutator group is a normal divisor that is different from and thus the whole group. ${\displaystyle \{1\}}$

The special linear group is perfect, but not easy. ${\displaystyle SL(2,\mathbb {F} _{5})}$

Web links

• Eric W. Weisstein : Perfect Group . In: MathWorld (English).