Multiplicative group of a field implies every finite subgroup is cyclic
This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., multiplicative group of a field) must also satisfy the second group property (i.e., group in which every finite subgroup is cyclic)
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(Note that the result holds more generally for the multiplicative group of an integral domain).
For finite fields
For infinite fields
- Classification of fields whose multiplicative group is locally cyclic
- Classification of fields whose multiplicative group is uniquely divisible
For commutative rings that are not fields
- Multiplicative group of a field implies at most n elements of order dividing n
- At most n elements of order dividing n implies every finite subgroup is cyclic
The proof follows directly by piecing together facts (1) and (2).