This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: normal subgroup and subgroup of finite index
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A subgroup of a group is said to be almost normal if it satisfies the following equivalent conditions:
- Its normalizer has finite index in the whole group.
- It is a normal subgroup of a subgroup of finite index in the whole group.
- It has only finitely many conjugate subgroups.
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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 subgroup property is a finitarily tautological subgroup property: when the ambient group is a finite group, the property is satisfied.
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This is a variation of normal subgroup|Find other variations of normal subgroup | Read a survey article on varying normal subgroup
Relation with other properties
Every subgroup of a group is almost normal if and only if the center has finite index, or equivalently, if the inner automorphism group of the group is finite.
- Groups with finite classes of conjugate subgroups by B.H. Neumann, Math. Z., 63, 1955, Pages 76-96