Fixed-point subgroup of a subgroup of the automorphism group implies local powering-invariant
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., fixed-point subgroup of a subgroup of the automorphism group) must also satisfy the second subgroup property (i.e., local powering-invariant subgroup)
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Suppose is a group. Suppose is a subgroup of the automorphism group of . Suppose is the subgroup of comprising precisely those elements that are fixed by every element of . In other words, is a fixed-point subgroup of a subgroup of the automorphism group in .
Then, is a local powering-invariant subgroup of : if and are such that there is a unique satisfying , then .
Given: Group , subgroup of . Subgroup of defined as the set of fixed points under of . and are such that there is a unique satisfying .
To prove: . In other words, for all .
Proof: We do the proof for fixed but arbitrary . We have that, since is an automorphism:
where the last step follows from the fact that and every element of is fixed by every automorphism in .
We thus obtain that . Since is the unique element whose power is , this forces , completing the proof.