Center not is intermediately 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., intermediately local powering-invariant subgroup)
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It is possible to have a solvable group such that the center is not an intermediately local powering-invariant subgroup of , i.e., there exists an intermediate subgroup of such that is not a local powering-invariant subgroup of .
Further information: amalgamated free product of Z and Z over 2Z
Consider the group , explicitly given as:
with the subgroup:
- The center of is the subgroup , which is the amalgamated copy of .
- is solvable: In fact, is isomorphic to the infinite dihedral group, which is a metacyclic group.
- The subgroup is isomorphic to , with living as inside it.
- is not local powering-invariant in : To see this, note that the element has unique square root but this square root is not in .