Center is quotient-local powering-invariant in nilpotent group
This article gives the statement, and possibly proof, of the fact that for any group, the subgroup obtained by applying a given subgroup-defining function (i.e., center) always satisfies a particular subgroup property (i.e., quotient-local powering-invariant subgroup)}
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Suppose is a nilpotent group and is the center of . Then, is a quotient-local powering-invariant subgroup of : if any element of has a unique root in for some prime number , then its image in has a unique root in .
- Center not is quotient-local powering-invariant in solvable group
- Second center not is local powering-invariant in solvable group
- Center is local powering-invariant
- Center is normal
- Local powering-invariant and normal iff quotient-local powering-invariant in nilpotent group
The proof follows directly by combining Facts (1), (2), and (3).