Local powering-invariant subgroup containing the center is intermediately powering-invariant in nilpotent group
Suppose is a nilpotent group and is a subgroup containing the center of that is also a local powering-invariant subgroup of . Then, is an intermediately powering-invariant subgroup of . Explicitly, suppose is a subgroup of containing . Then, is a powering-invariant subgroup of .
- Local powering-invariant subgroup containing the center is intermediately local powering-invariant in nilpotent group
- Torsion-freeness for a prime is subgroup-closed
- Equivalence of definitions of nilpotent group that is torsion-free for a set of primes
Given: A nilpotent group , a subgroup of that is local powering-invariant and such that where is the center of . A subgroup of containing . A prime number such that is -powered. An element .
To prove: There exists a unique element such that .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|2||is -torsion-free.||is -powered (we are using the powering-injectivity here)||given-direct|
|3||is -torsion-free.||Fact (1)||Steps (1), (2)||Step-fact combination direct|
|4||The map is injective in .||Fact (2)||is nilpotent||Step (3)||Step-fact combination direct (specifically, we want to use the implication from (4) to (1) in the multi-part equivalence of Fact (2))|
|5||There exists an element satisfying .||is -powered (we are using the -divisibility here),||given-direct|
|6||The element of the preceding step is the unique root of in all of .||Steps (4), (5)||Step-combination direct|
|7||The element of Step (5) is in .||is local powering-invariant in .||Step (6)||Step-given combination direct.|