Second center not is 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., second center) need not satisfy a particular subgroup property (i.e., local powering-invariant subgroup)
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It is possible to have a solvable group such that the second center of is not a local powering-invariant subgroup of . In other words, there exists an element in the second center and a natural number such that the element has a unique root in but this root is not inside the second center.
- Center is local powering-invariant
- Upper central series members are local powering-invariant in nilpotent group
Define (here 1 denotes the identity element):
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.
- is in the second center, based on the description of the second center as .
- 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 .
- The unique square root of is not in the second center: This follows from the explicit description of the second center and the information above.