Characteristic subgroup of abelian group is quotient-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., characteristic subgroup of abelian group) must also satisfy the second subgroup property (i.e., quotient-powering-invariant subgroup)
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Suppose is an abelian group and is a characteristic subgroup of (in other words, is a characteristic subgroup of abelian group). Then, is a quotient-powering-invariant subgroup of : if is powered over a prime (i.e., every element of has a unique root), so is the quotient group .
- Characteristic subgroup of abelian group is powering-invariant
- Powering-invariant and central implies quotient-powering-invariant
The proof follows directly from Facts (1) and (2).