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Semisimple group

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

Contents

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

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
simple non-Abelian group
quasisimple group
characteristically simple group

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
perfect group equals its own derived subgroup semisimple implies perfect perfect not implies semisimple

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.