# Left-transitively complemented normal 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 **left-transitively complemented normal subgroup** if whenever the whole group is a complemented normal subgroup of a bigger group, the subgroup is also a complemented normal subgroup of that group.

### Definition with symbols

A subgroup of a group is termed a **left-transitively complemented normal subgroup** if, for any group containing such that is a complemented normal subgroup of , then is also a complemented normal subgroup of .

## Formalisms

### In terms of the left transiter

This property is obtained by applying the left transiter to the property: complemented normal subgroup

View other properties obtained by applying the left transiter

## Relation with other properties

### Stronger properties

- Sylow direct factor
- Hall direct factor
- Characteristically complemented characteristic subgroup
- Complete characteristic direct factor
- Complemented normal subgroup of complete characteristic direct factor

### Weaker properties

- Characteristic subgroup:
*For proof of the implication, refer Left-transitively complemented normal implies characteristic and for proof of its strictness (i.e. the reverse implication being false) refer Characteristic not implies left-transitively complemented normal*. - Complemented characteristic subgroup:
*For proof of the implication, refer Left-transitively complemented normal implies complemented characteristic and for proof of its strictness (i.e. the reverse implication being false) refer Complemented characteristic not implies left-transitively complemented normal*. - Complemented normal subgroup