Nilpotent-quotient implies subgroup-to-quotient powering-invariance implication
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., nilpotent-quotient subgroup) must also satisfy the second subgroup property (i.e., normal subgroup satisfying the subgroup-to-quotient powering-invariance implication)
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Suppose is a group and is a normal subgroup of that is a nilpotent-quotient subgroup, i.e., the quotient group is a nilpotent group (note that and themselves may or may not be nilpotent). Then, is a normal subgroup satisfying the subgroup-to-quotient powering-invariance implication in . Explicitly, this means that if is a prime number such that both and are -powered, then so is .
- Divisibility is inherited by quotient groups
- Equivalence of definitions of nilpotent group that is torsion-free for a set of primes: We use the equivalence of (1) and (2) within the multi-part equivalence. This says that a nilpotent group has no non-identity elements of order if and only if its power map is injective.
Given: Group , normal subgroup such that both and are -powered for some prime number .
To prove: is -powered.
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||is -divisible, i.e., any element in it has a root. In other words, the power map in it is surjective.||Fact (1)||is -powered, hence -divisible, is normal in .|
|2||is -torsion-free.||and are both -powered.||Any element of has a unique root in , and also a unique root in , so those unique roots coincide, which means it has no root outside . Thus, has no non-identity element of order .|
|3||The power map in is injective.||Fact (2)||is nilpotent.||Step (2)||Fact-given-step combination direct.|
|4||is -powered, i.e., its power map is bijective.||Steps (1), (3)||Step-combination direct.|