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Wedderburn's little theorem : every finite division ring $D$ is commutative, or $D^*$ is abelian group.

Now if $D^*$ be a finitely generated group then $D^*$ is an abelian group ?

Eric Wofsey
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amir bahadory
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  • Just to be sure, when $D$ is a division ring, does $D^*$ infinite and finitely generated ever happen (abelian or not)? Nice question by the way. – Clément Guérin May 12 '18 at 07:00

1 Answers1

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This isn't really an answer, but it looks as though the question might be hard.

There's a proof for the case where $D$ is finitely generated over its centre in Theorem 1 of

Akbari, S.; Mahdavi-Hezavehi, M., Normal subgroups of $\text{GL}_n(D)$ are not finitely generated, Proc. Am. Math. Soc. 128, No.6, 1627-1632 (2000). ZBL0951.20036,

and Conjecture 1 in the same paper is that it's true in general.

Incidentally, it also seems to be an open problem to decide whether an infinite division ring can be finitely generated as a ring (which of course would follow if its group of units were finitely generated as a group). This is referred to as the "Latyshev problem" in these slides of a 2014 talk by Agata Smoktunovicz.

Jeremy Rickard
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