# Group having no proper nontrivial transitively normal subgroup

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## Definition

A nontrivial group is termed a **group having no proper nontrivial transitively normal subgroup** if the only transitively normal subgroups of the group are the whole group and the trivial subgroup. Here, a subgroup is transitively normal if every normal subgroup of the subgroup is normal in the whole group.

## Formalisms

### In terms of the simple group operator

This property is obtained by applying the simple group operator to the property: transitively normal subgroup

View other properties obtained by applying the simple group operator

## Relation with other properties

### Stronger properties

- Simple group:
*For proof of the implication, refer Simple implies no proper nontrivial transitively normal subgroup and for proof of its strictness (i.e. the reverse implication being false) refer No proper nontrivial transitively normal subgroup not implies simple*.

### Weaker properties

- Centrally indecomposable group
- Directly indecomposable group
- Centerless group unless the group is a simple Abelian group, i.e., a cyclic group of prime order.