Weakly pronormal 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]
This is a variation of normality|Find other variations of normality | Read a survey article on varying normality

Definition

Definition with symbols

A subgroup H of a group G is termed weakly pronormal or is said to satisfy the Frattini property if it satisfies the following equivalent conditions:

  • Given any g \in G, there exists x \in H^{\langle g \rangle} such that H^x = H^g. Here H^{\langle g \rangle} denotes the smallest subgroup of G containing H which is closed under conjugation by g.
  • If H \le K \le L \le G are such that K is a normal subgroup of L, we have KN_L(H) = L.

Equivalence of definitions

Further information: Equivalence of definitions of weakly pronormal subgroup

Relation with other properties

For a picture of related subnormal-to-normal subgroup properties, refer this implication diagram.

Stronger properties

Weaker properties

References

Journal references