# Subnormal not implies normal

From Groupprops

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., subnormal subgroup) neednotsatisfy the second subgroup property (i.e., normal subgroup)

View a complete list of subgroup property non-implications | View a complete list of subgroup property implications

Get more facts about subnormal subgroup|Get more facts about normal subgroup

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property subnormal subgroup but not normal subgroup|View examples of subgroups satisfying property subnormal subgroup and normal subgroup

## Statement

A subnormal subgroup of a group need not be normal.

## Partial truth

A group in which every subnormal subgroup is normal is termed a T-group.

## Proof

This statement is easily shown to reduce to the statement that normality is not transitive, i.e. a normal subgroup of a normal subgroup need not be normal.

`Further information: Normality is not transitive`