Divisibility is central extension-closed
This article gives the statement, and possibly proof, of a group property (i.e., divisible group for a set of primes) satisfying a group metaproperty (i.e., central extension-closed group property)
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- is -divisible.
- The quotient group is -divisible.
Then, the whole group is -divisible.
The dual fact is that powering-injectivity is inherited by central extensions.
- Powering is central extension-closed
- Powering-injectivity is central extension-closed
- Divisibility is inherited by extensions where the normal subgroup is contained in the hypercenter
The idea is that of successive approximation. We first obtain a root in the quotient group, then pick a representative. We then pick a representative, and measure the extent to which its power misses the mark. Then, we take a root of that, and use that as the "first-order correction" to our original choice of representative.
The subgroup being central is crucial in making sure that the product of the powers of the original representative and the correction is the power of the product.
Given: A group , a central subgroup of , a prime such that both and are -divisible. An element .
To prove: There exists such that .
Proof: Suppose is the quotient map.
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation||Commentary|
|1||Let , so .||moved to quotient group|
|2||There exists such that .||is -divisible.||Step (1)||direct||took root in quotient group|
|3||Suppose , i.e., is such that . Then, is an element of .||Steps (1), (2)||We have that which is the identity element of . Thus, .||picked representative, this is our "first guess" , and measured how far it is, using .|
|4||There exists such that .||is -divisible.||Step (3)||direct||picked as the "correction" for our first guess.|
|5||The element works, i.e., .||is central.||Steps (3), (4)||We have because and is central. Thus, , as desired.||multiplied the first guess and the correction to get a correct guess.|