Local powering-invariant and normal iff quotient-local powering-invariant in nilpotent group
- is a local powering-invariant normal subgroup of , i.e., is both a local powering-invariant subgroup of and a normal subgroup of .
- is a quotient-local powering-invariant subgroup of .
- Local powering-invariant and normal not implies quotient-local powering-invariant
- Center not is quotient-local powering-invariant
- Local powering-invariance is quotient-transitive in nilpotent group
- Upper central series members are local powering-invariant in nilpotent group
- Quotient-local powering-invariant implies local powering-invariant
- Equivalence of definitions of nilpotent group that is torsion-free for a set of primes
- Nilpotency is quotient-closed
(2) implies (1)
This follows from Fact (1) (note that normality is definitional).
(1) implies (2)
Given: A nilpotent group , a subgroup of . is normal and local powering-invariant in . An element and a prime number such that the equation has a unique solution . is the quotient map.
To prove: is the unique root of in the quotient group .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||The map is injective from to itself.||Fact (2) (implication (3) implies (1) in the equivalence on the fact page)||is nilpotent and has a unique root.||Step-fact combination direct.|
|2||For any satisfying , we have .||is local powering-invariant in .||Step (1)||Let . By Step (1), the -power map is injective in , hence is the unique root of in . Since is local powering-invariant, this forces .|
|3||is -torsion-free.||Step (2)||If an element of has order , any inverse image of it in is an element outside whose power is in , contradicting Step (2).|
|4||is nilpotent.||Fact (3)||is nilpotent||Given-fact direct|
|5||The map is injective from to itself.||Fact (2)||Steps (3), (4)||Direct by combining the steps and the fact.|
|6||is the unique root of in .||Step (5)||Applying to gives that . The injectivity of the map by Step (5) now gives uniqueness.|