Abelian direct factor implies potentially verbal in finite
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a Finite group (?), every subgroup satisfying the first subgroup property (i.e., Abelian direct factor (?)) must also satisfy the second subgroup property (i.e., Potentially verbal subgroup (?)). In other words, every Abelian direct factor of finite group is a potentially verbal subgroup of finite group.
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- Central implies potentially verbal in finite: The proof idea is the same,but it uses a central product in place of a direct product.
Given: A finite group with direct factor . Let be a normal complement to in , so .
To prove: There exists a group containing such that is a verbal subgroup of .
Proof: If is trivial, we can set and be done. So, we can assume is nontrivial.
- By fact (1), we can write as a direct product of cyclic groups, say , where each is cyclic of prime power order, say .
- For each prime , let be a positive integer greater than the largest power of dividing the order of . (Note that the same may repeat among multiple s).
- Now consider the group where each is cyclic of order . Treat as the subgroup of obtained by embedding each naturally into the corresponding .
- Define , with embedded in using the embedding of in provided above.
- Let where the product is over the primes dividing the order of . Then, it is clear that the set of powers in is precisely . In particular, is a verbal subgroup of .