Centerless 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
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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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Definition
Symbol-free definition
A group is said to be centerless if it satisfies the following equivalent conditions:
- The center of the group is trivial.
- The natural homomorphism from the group to its automorphism group that sends each element to the conjugation it induces, is injective (viz no two elements induce the same inner automorphism)
Definition with symbols
A group is said to be centerless if it satisfies the following equivalent conditions:
- The center
is the trivial group.
- The natural homomorphism
given by
, where
, is an injective map from
to
.
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
simple non-abelian group | non-abelian and simple: has no proper nontrivial normal subgroup | simple and non-abelian implies centerless | symmetric group:S3 is a centerless group that is not simple. | |FULL LIST, MORE INFO |
almost simple group | in between a simple non-abelian group and its automorphism group | almost simple implies centerless | symmetric group:S3 is a centerless group that is not almost simple. | |FULL LIST, MORE INFO |
complete group | centerless and every automorphism is inner | (by definition) | alternating group:A4 is centerless but not complete, it has outer automorphisms realized as conjugations via its embedding as A4 in S4. | |FULL LIST, MORE INFO |
characteristically simple non-abelian group | non-abelian and characteristically simple: has no proper nontrivial characteristic subgroup | characteristically simple and non-abelian implies centerless | symmetric group:S3 is a centerless group that is not simple. | |FULL LIST, MORE INFO |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
capable group | can be expressed as (up to isomorphism) the inner automorphism group of some group; equivalently, the epicenter is trivial. | centerless implies capable -- basically, it is isomorphic to its own inner automorphism group | |FULL LIST, MORE INFO | |
stem group | the center is contained in the derived subgroup. | |FULL LIST, MORE INFO | ||
unicentral group | epicenter equals center. | |FULL LIST, MORE INFO |
Metaproperties
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 centerless groups is centerless. This follows from the general fact that the center of a direct product equals the direct product of the individual centers.