Descendance does not satisfy image condition
This article gives the statement, and possibly proof, of a subgroup property (i.e., descendant subgroup) not satisfying a subgroup metaproperty (i.e., image condition).
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It is possible to have groups and , a descendant subgroup of and a surjective homomorphism such that is not a descendant subgroup of .
Example of the infinite dihedral group
- 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: