Subnormal not implies normal

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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) need not satisfy 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