Division Rings, Finite Subgroups are Cyclic

Finite Subgroups are Cyclic

Let G be a finite, multiplicative subgroup in the division ring R. We will show that G is cyclic, but only if R has characteristic p. In the quaternions, a ring with characteristic 0, G might be ±(1,i,j,k). This multiplicative subgroup isn't even abelian, much less cyclic.

Returning to characteristic p, let G be a finite multiplicative group with identity e. Remember that e is synonymous with 1 in R.

Let W be the set of linear combinations of group elements with integer coefficients. A typical element of W might look like 7G₁+11G₅+2G₉. Of course 2G₉ is shorthand for G₉+G₉. show that W forms a ring inside R.

Since the coefficients are bounded by p, and since G is a finite group, W is a finite subring of R. Since R is a division ring, W is a finite domain. By the previous theorem, W is a field. The nonzero elements of a finite field form a cyclic multiplicative group, and G is a subgroup of F^*, hence G is cyclic.

Note, we only need R to be a domain for this to work.