# SCDIN-subgroup

From Groupprops

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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 of a group is termed a **SCDIN-subgroup**, or **subset-conjugacy-determined in normalizer**, if, whenever are subsets of such that there exists with , there exists such that for all .

## 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 |