# Descendance does not satisfy image condition

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., descendant subgroup)notsatisfying a subgroup metaproperty (i.e., image condition).

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## Statement

It is possible to have groups and , a descendant subgroup of and a surjective homomorphism such that is not a descendant subgroup of .

## Proof

### Example of the infinite dihedral group

We take:

- to be the infinite dihedral group. We can take the presentation .
- to be a subgroup of order two complementary to the cyclic maximal subgroup, i.e., is a subgroup generated by a
*reflection*. In the above presentation, we can take . - to be the quotient of by the subgroup generated by multiples of 3 in the cyclic maximal subgroup, with the natural quotient map. In symbols, . is isomorphic to symmetric group:S3, and the image of under the quotient map corresponds to S2 in S3.

We can check that:

- is descendant in : It is the intersection of the following descending chain of subgroups, each of index two in its predecessor, hence each normal in its predecessor:

- The image of in is not descendant in : The image looks like S2 in S3, which is a subgroup of a finite group, and is in fact a contranormal subgroup, so it cannot be descendant.