Groupprops, The Group Properties Wiki (pre-alpha)

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
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RANDOM GROUP PROPERTY: T-group: A group where any subnormal subgroup is normal, i.e., where normality is transitive. In general, normality is not transitive.

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