Powering-invariance is union-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., 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 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 powering-invariant subgroup of .
Given: is a group and is a collection of powering-invariant subgroups of . The union is a subgroup of . is powered over a prime . An element .
To prove: There exists such that .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||There exists such that .||is the union of , .||Follows directly from given.|
|2||is -powered.||is -powered, is powering-invariant in .||Directly from given data used.|
|3||There exists such that .||Steps (1), (2)||direct from steps.|
|4||There exists such that .||is the union of||Step (3)||Step-given direct.|