P-automorphism-invariant 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

Let p be a prime number. A subgroup H of a p-group G (i.e., a group where the order of every element is a power of p) is termed p-automorphism-invariant if, for any automorphism \sigma of G such that the order of \sigma is a power of p, \sigma(H) = H.

For the property the context of finite p-groups, refer p-automorphism-invariant subgroup of finite p-group.

Relation with other properties

Equivalent properties

For a finite p-group, the property of being p-automorphism invariant equals the property of being a subnormal stability automorphism-invariant subgroup. That's because any p-automorphism can be realized as the stability automorphism of a subnormal series, and conversely, any stability automorphism of a subnormal series must be a p-automorphism. Further information: Stability group of subnormal series of p-group is p-group, p-group of automorphisms of p-group is contained in stability group of some normal series

Stronger properties

Weaker properties