Equivalence of definitions of local powering-invariant subgroup
This article gives a proof/explanation of the equivalence of multiple definitions for the term local powering-invariant subgroup
View a complete list of pages giving proofs of equivalence of definitions
The following are equivalent for a subgroup of a group :
- Whenever and are such that there is a unique such that , we must have .
- Whenever and is a prime number such that there is a unique such that , we must have .
(1) implies (2)
This is obvious.
(2) implies (1)
We use Fact (1) to show uniqueness of root for some prime dividing . Then, use condition (2) to obtain that that root is also in . Now repeat the process with this new element -- we're now finding a root of the element. Each step gets rid of one prime divisor.