Quotient-powering-invariance is union-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., quotient-powering-invariant subgroup) satisfying a subgroup metaproperty (i.e., union-closed subgroup property)
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Suppose is a group and is a collection of quotient-powering-invariant subgroups of . Suppose the union is a subgroup of (and hence is also the same as the join of subgroups . Then, this union of also a quotient-powering-invariant subgroup of .
Note that existence of roots is guaranteed (see divisibility is inherited by quotient groups). The part we need to establish is uniqueness. Note also that by Fact (1), the union must be a normal subgroup if it is a subgroup, hence we will assume this in our setup.
Given: is a group and is a collection of quotient-powering-invariant subgroups of . The union is a subgroup of . is powered over a prime . Two elements that are in the same coset of (concretely, ). are respectively the unique solutions to and .
To prove: and are in the same coset of .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||There exists such that . In other words, there exists such that and are in the same coset of .||is the union of , .||Follows directly from given.|
|2||and are in the same coset of .||is -powered, is quotient-powering-invariant in .||Step (1)||Direct from given and Step (1). Note that quotient-powering-invariant in a -powered group means that the roots of elements in the same coset are in the same coset.|
|3||and are in the same coset of .||is the union of .||Step (2)||Since , being in the same coset of implies being in the same coset of .|