Direct factor of characteristic subgroup

This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: direct factor and characteristic subgroup
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Definition

Symbol-free definition

A subgroup of a group is termed a direct factor of characteristic subgroup if it satisfies the following equivalent conditions:

• It can be expressed as a direct factor of a characteristic subgroup
• It is a direct factor in its characteristic closure

In terms of the composition operator

The subgroup property of being a direct factor of characteristic subgroup is obtained by applying the composition operator to the subgroup properties of being a direct factor and of being characteristic.

Relation with other properties

Stronger properties

• Characteristic subgroup
• Direct factor
• Minimal normal subgroup
• Direct factor of fully characteristic subgroup
• Direct root of characteristic subgroup

Weaker properties

• Central factor of characteristic subgroup
• Direct factor of normal subgroup
• Normal subgroup of characteristic subgroup
• 2-subnormal subgroup
• Subnormal subgroup

Related properties

• Characteristic subgroup of direct factor