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]

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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 $ainA$.

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