Frattini-embedded normal subgroup

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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, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

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, NH \ne 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