Center not is quotient-local powering-invariant in solvable group
This article gives the statement, and possibly proof, of the fact that in a group satisfying the property solvable group, the subgroup obtained by applying a given subgroup-defining function (i.e., center) need not satisfy a particular subgroup property (i.e., quotient-local powering-invariant subgroup)
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It is possible to have a solvable group with center such that is not a quotient-local powering-invariant subgroup of . Explicitly, let be the quotient map. Then, there exists a prime number and an element such that has a unique root in , but has multiple roots in .
We can understand the structure of using the following normal series:
The successive quotients are . More details below:
- is the center and the quotient group is isomorphic to the amalgamated free product , with the two pieces generated by the images of and and the amalgamated part being the image of , which coincides with the image of .
- is the second center and the quotient group is isomorphic to the free product , which in turn is isomorphic to the infinite dihedral group (where the images of and are both reflections whose product gives a generator for the cyclic maximal subgroup).
- is the cyclic maximal subgroup inside .
Consider now the element . We have the following:
- is solvable: This is obvious from the normal series for where all the quotients are abelian.
- has a unique square root, namely , in : This requires some work to show rigorously, and can be demonstrated using a polycyclic presentation with the elements where . The idea is to compute the general expression for the square of an arbitrary element that is of the form and deduce that, for the square to equal , we must have .
- Under the quotient map by the center, the image of does not have a unique square root: The center is , and the quotient group is . As noted above, the image of coincides with the image of in the quotient, but the images of and do not coincide. Thus, the image of modulo the center does not have a unique square root in the quotient group by the center.