SCDIN-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

Symbol-free definition

A subgroup of a group is termed a SCDIN-subgroup, or subset-conjugacy-determined in normalizer, if it is a subset-conjugacy-determined subgroup inside its normalizer, relative to the whole group.

Definition

A subgroup H of a group G is termed a SCDIN-subgroup, or subset-conjugacy-determined in normalizer, if, whenever A,B are subsets of H such that there exists g \in G with gAg^{-1} =B, there exists k \in N_G(H) such that kak^{-1} = gag^{-1} for all a 
in A.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
direct factor
central factor
retract
subset-conjugacy-closed subgroup subset-conjugacy-determined in itself
normal subgroup
Sylow TI-subgroup Sylow and TI implies SCDIN
Abelian pronormal subgroup Abelian and pronormal implies SCDIN
Abelian Sylow subgroup Abelian and Sylow implies SCDIN

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
NSCDIN-subgroup
CDIN-subgroup
WSCDIN-subgroup
WNSCDIN-subgroup
MWNSCDIN-subgroup