Left-transitively complemented normal 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 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 $H$ of a group $G$ is termed a left-transitively complemented normal subgroup if, for any group $K$ containing $G$ such that $G$ is a complemented normal subgroup of $K$, then $H$ is also a complemented normal subgroup of $K$.

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

Related properties

• Right-transitively complemented normal subgroup