Local powering-invariant subgroup containing the center is intermediately local 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 local powering-invariant subgroup of . Explicitly, suppose is a subgroup of containing . Then, is a local powering-invariant subgroup of .
- Local powering-invariant subgroup containing the center is intermediately 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
- Nilpotency is subgroup-closed
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 and an element such that there is a unique element satisfying .
To prove: There exists a unique element such that .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|2||is nilpotent.||Fact (3)||is nilpotent,||Given-fact direct|
|3||is -torsion-free.||Fact (2)||has a unique root in .||We use the equivalence (3) implies (1) within the multi-part equivalence of Fact (2).|
|4||is -torsion-free.||Fact (1)||Steps (1), (3)||Step-fact combination direct|
|5||The map is injective in .||Fact (2)||is nilpotent||Step (4)||Step-fact combination direct (specifically, we want to use the implication from (4) to (1) in the multi-part equivalence of Fact (2))|
|6||The element is the unique root of in all of .||satisfies .||Step (5)||given-step direct|
|7||The element of Step (6) is in .||is local powering-invariant in .||Step (6)||Step-given combination direct.|