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
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RANDOM GROUP PROPERTY: Group satisfying normalizer condition: A group with no proper self-normalizing subgroup.


Contents

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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View a complete list of semi-basic definitions on this wiki

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 G is said to be centerless if it satisfies the following equivalent conditions:

  • The center Z(G) is the trivial group.
  • The natural homomorphism G \to Aut(G) given by g \mapsto c_g, where c_g = x \mapsto gxg^{-1}, is an injective map from G to Aut(G).

Relation with other properties

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.

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