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Frattini-embedded normal subgroup
From Groupprops
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: 2-subnormal subgroup: A normal subgroup of a normal subgroup. Need not be normal, because normality is not transitive.
Contents |
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
