Powering-invariant over quotient-powering-invariant implies powering-invariant
Suppose is a group and are subgroups of such that , is normal in , and the following conditions hold:
Then, is a powering-invariant subgroup of .
Given: is a group and are subgroups of such that , is normal in , and the following conditions hold:
- is a quotient-powering-invariant subgroup of .
- is a powering-invariant subgroup of .
is a prime number such that is powered over . An element .
To prove: There exists such that .
Proof: Let be the quotient map.
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||There exists such that .||is -powered.||Given-direct.|
|2||is -powered.||is quotient-powering-invariant in , is -powered.||Given-direct.|
|3||and is the unique root of in .||Steps (1), (2)||Applying the homomorphism to Step (1) gives that . Since, by Step (2), is -powered, must be the unique root.|
|4||.||is powering-invariant in .||Steps (2), (3)||By Step (2), is -powered, hence is -powered from the given information. Thus, the unique root of in , which equals ,(from Step (3)) must be in .|
|5||.||Step (4)||Since , , which is since .|