Frattini-embedded normal subgroup

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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