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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof.
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RANDOM SUBGROUP PROPERTY: Conjugacy-closed subgroup: A subgroup such that any two elements of the subgroup that are conjugate in the whole group are conjugate in the subgroup.

Contents 1 Definition 1.1 Definition with symbols 2 Effect of property operators 2.1 Left transiter 3 References

Definition

Definition with symbols

A normal subgroup N of a group G is termed Frattini-embedded or Frattini-imbedded if for every proper subgroup H of G, . When G is a finite group, or more generally, when every proper subgroup of G is contained in a maximal subgroup, then this condition is equivalent to saying that N is contained in the Frattini subgroup of G.

Effect of property operators

Left transiter

Any characteristic subgroup of a Frattini-embedded normal subgroup is Frattini-embedded normal. It's not clear whether characteristicity is precisely the left transiter of the property of being Frattini-embedded normal.

References

• An Essay on Frattini Imbedded Normal Subgroups by R. Baer, Comm. Pure Appl. Math. 26, 609--658 (1973)
• On Frattini Imbedded Normal Subgroups by Martin Newell