# Index two not implies characteristic

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., subgroup of index two) neednotsatisfy the second subgroup property (i.e., characteristic subgroup)

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

Get more facts about subgroup of index two|Get more facts about characteristic subgroup

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property subgroup of index two but not characteristic subgroup|View examples of subgroups satisfying property subgroup of index two and characteristic subgroup

## Statement

A subgroup of index two in a group need not be a characteristic subgroup.

## Proof

### Example of the Klein four-group

`Further information: Klein four-group`