Periodicity is extension-closed
This article gives the statement, and possibly proof, of a group property (i.e., periodic group) satisfying a group metaproperty (i.e., extension-closed group property)
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Further, if is a set of primes such that both and are -groups -- all prime factors of orders of elements in these groups are in -- then is also a -group.
We prove the second version. Note that taking to be the set of all primes will give the first version.
Given: A group . A normal subgroup with quotient group . A prime set such that both and are -groups: all elements of and all elements of have finite orders with all prime divisors of the order of every element in .
To prove: Every element of has finite order and every prime divisor of the order of every element is in .
Proof: We pick an arbitrary element . We let be the quotient map.
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||has order equal to some -number as an element of .||is a -group.|
|2||where is the -number chosen in Step (1).||Step (1)|
|3||Let from Step (2). Then, has order equal to some -number as an element of .||is a -group||Step (2)|
|4||is the identity element of . Thus, the order of in is finite and divides the -number , hence it must also be a -number.||Step (3)|