Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Finite Groups Embeddable in Division Rings   Finite groups that are embeddable in the multiplicative groups of division
rings $K$ were completely determined by S. A. Amitsur in 1955. In case $K$ has
characteristic $p>0$, the only possible finite subgroups of $K^*$ are cyclic
groups, according to a theorem of I. N. Herstein. Thus, the only interesting
case is when $K$ has characteristic 0; that is, when $K\supseteq {\Bbb Q}$.
  Herstein conjectured that odd-order subgroups of division rings $K$ were
cyclic, and he proved this to be the case when $K$ is the division ring of the
real quaternions. Herstein's conjecture was settled negatively by Amitsur. As
part of his complete classification of finite groups in division rings, Amitsur
showed that the smallest noncyclic odd-order group that can be embedded in a
division ring is one of order 63 (and this group is unique).
  Amitsur's paper is daunting to read as it is long and technically
complicated. In lecturing to a graduate class on division rings, I tried to
find a simple reason for the ``first exceptional odd order'' 63 (to Herstein's
conjecture). After some work, I did come up with a reason that was simple
enough to be explained to my class, without having to go through any part of
Amitsur's paper. Furthermore, the method I used led easily to the second
exceptional odd order, 117 (which was not mentioned in Amitsur's paper). Since
this line of reasoning did not seem to have appeared in the literature before,
I record it in this short note. To better motivate the results discussed here,
I have also included a quick exposition on the beginning part of the theory of
finite subgroups of division rings.

8
The categories of this paper are: hep-ex
7
The categories of this paper are: hep-th
9
The categories of this paper are: nlin
37
The categories of this paper are: supr-con
19
The categories of this paper are: eess
21
The categories of this paper are: adap-org
6
The categories of this paper are: astro-ph
3
The categories of this paper are: physics
12
The categories of this paper are: nucl-th
36
The categories of this paper are: cmp-lg
15
The categories of this paper are: nucl-ex
0
The categories of this paper are: hep-ph
13
The categories of this paper are: quant-ph
31
The categories of this paper are: ao-sci
27
The categories of this paper are: comp-gas
22
The categories of this paper are: chao-dyn