Simple non-abelian group

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This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: simple group and non-Abelian group
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This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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This article is about a term related to the Classification of finite simple groups

Definition

Symbol-free definition

A group is said to be a simple non-abelian group if:

  • It is simple, i.e., it has no proper nontrivial normal subgroups
  • It is not abelian, i.e., it is not true that any two elements in the group commute.

Facts

Every subgroup-defining function gives trivial group or whole group

Since any subgroup-defining function (such as the center, the commutator subgroup, the Frattini subgroup, the Fitting subgroup etc.) returns a characteristic subgroup of the whole group, and since every characteristic subgroup is normal, any subgroup obtained via a subgroup-defining function must be either trivial or the whole group. This, combined with the fact that the group is non-Abelian, tells us the following:

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
infinite simple group infinite and a simple group the only simple abelian groups are the groups of prime order, which are all finite. there are finite simple non-abelian groups
finite simple non-abelian group finite, simple, and non-abelian direct there are infinite simple groups, which are hence non-abelian.

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
perfect group equals its own derived subgroup simple and non-abelian implies perfect SL(2,5) is an example of a perfect group that is not simple. Quasisimple group, Semisimple group|FULL LIST, MORE INFO
centerless group its center is a trivial group simple and non-abelian implies centerless symmetric group:S3 is a centerless group that is not simple. |FULL LIST, MORE INFO
quasisimple group perfect group whose inner automorphism group is simple non-abelian. simple and non-abelian implies quasisimple SL(2,5), and more generally, any universal central extension of a simple non-abelian group that is not Schur-trivial, gives an example. |FULL LIST, MORE INFO
semisimple group central product of quasisimple groups. (via quasisimple) (via quasisimple) |FULL LIST, MORE INFO
almost simple group can be embedded between a simple non-abelian group and its automorphism group (by definition) symmetric group:S5 is an almost simple group that is not simple. More generally, the automorphism group of any simple non-abelian group that is not complete. |FULL LIST, MORE INFO
capable group can be expressed as the inner automorphism group of some group. (via centerless) (via centerless) Centerless group|FULL LIST, MORE INFO