From Groupprops
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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
VIEW RELATED:
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
Weaker properties
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