Groupprops, The Group Properties Wiki (pre-alpha)

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From Groupprops

Contents 1 Definition 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties 3 Metaproperties 3.1 Subgroups 3.2 Quotients 3.3 Characteristic subgroups

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This article defines a group property: a property that can be evaluated to true/false for any given group
View a complete list of group properties
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RANDOM GROUP PROPERTY: Complete group: A group for which the natural homomorphism to its automorphism group is an isomorphism, i.e., a centerless group for which every automorphism is inner.

This term is related to the problem of realization related to the following subgroup-defining function: Frattini subgroup
Realization problems are usually about which groups can be realized as subgroups/quotients related to a subgroup-defining function.
View other terminology related to realization problems for Frattini subgroup OR View facts related to them

Definition

A group N is termed Frattini-embedded normal-realizable if there exists a group G and an embedding of N in G such that N is a Frattini-embedded normal subgroup of G. In other words, N is a normal subgroup and NH is a proper subgroup of G for any proper subgroup H of G.

Relation with other properties

Stronger properties

• Finite-Frattini-realizable group
• Finite-(Frattini-embedded normal)-realizable group

Weaker properties

• Inner-in-automorphism-Frattini group: This condition states that the inner automorphism group lies inside the Frattini subgroup of the automorphism group. For full proof, refer: Frattini-embedded normal-realizable implies inner-in-automorphism-Frattini
• ACIC-group: For full proof, refer: Frattini-embedded normal-realizable implies ACIC

Metaproperties

Subgroups

This group property is not subgroup-closed, viz., we can have a group satisfying the property, with a subgroup not satisfying the property

Quotients

This group property is not quotient-closed, viz., we could have a group with the property and a quotient group of that group that does not have the property

Characteristic subgroups

This group property is characteristic subgroup-closed: any characteristic subgroup of a group with the property, also has the property