Characteristicity does not satisfy image condition
This article gives the statement, and possibly proof, of a subgroup property (i.e., characteristic subgroup) not satisfying a subgroup metaproperty (i.e., image condition).
View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about characteristic subgroup|Get more facts about image condition|
Statement with symbols
If is a characteristic subgroup of and is a surjective homomorphism, then is not necessarily a characteristic subgroup of .
Although the image of every characteristic subgroup under a surjective homomorphism need not be characteristic, there are special situations where we can guarantee this property. A characteristic subgroup whose image under every surjective homomorphism is characteristic is termed an image-closed characteristic subgroup.
It turns out that every verbal subgroup, and in particular, every member of the lower central series and derived series, as well as the agemo subgroups, are image-closed characteristic. In fact, their images are verbal subgroup with the same defining words.
We present here two counterexamples of groups of order eight.
Example of an Abelian group of order eight
Let , and be the subgroup . This is (the first omega subgroup) -- the set of elements of order two in . Thus, is a characteristic subgroup of .
Let , and be the quotient map. Then, is isomorphic to the Klein-four group, and is a subgroup of order two in it, which is clearly not characteristic in .
Example of the dihedral group
Consider the dihedral group of order eight with an element of order four, playing the role of the rotation, and an element of order two, playing the role of the reflection. Let be the cyclic subgroup of order four generated by . Then, is a characteristic subgroup of .
Consider the quotient map where . The group is isomorphic to a Klein-four group, and the image is a subgroup of order two in that Klein-four group. Clearly, is not characteristic inside .