Semisimple group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
This is a variation of simplicity|Find other variations of simplicity | Read a survey article on varying simplicity

The term semisimple has also been used at some places for a group whose solvable radical is trivial, which is equivalent to being a Fitting-free group

Definition

Symbol-free definition

A group is said to be semisimple if it occurs as a central product of (possibly more than two) quasisimple groups.

Definition with symbols

A group G is said to be semisimple if there are subgroups G_1, G_2, \ldots, G_r such that:

  • Each G_i is quasisimple
  • The G_is generate G
  • The group [G_i, G_j] is trivial for all i \ne j

Relation with other properties

Stronger properties

Weaker properties

Metaproperties

Template:S-universal

Every finite group can be realized as a subgroup of a semisimple group. This follows from the fact that every finite group can be realized as a subgroup of a simple non-Abelian group.

Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
View a complete list of quotient-closed group properties

Every quotient of a semisimple group is semisimple. This follows from the fact that every quotient of a quasisimple group is quasisimple, and that a central product is preserved on going to the quotient.

Direct products

This group property is direct product-closed, viz., the direct product of an arbitrary (possibly infinite) family of groups each having the property, also has the property
View other direct product-closed group properties

A direct product of semisimple groups is semisimple. In fact, any central product of semisimple groups is semisimple.