View source for Multiplicative group

{{TEX|done}}{{MSC|12E15|}}

''of a skew-field''

The group of all elements of the given [[Skew-field|skew-field]] except the zero element and with the operation of multiplication in the skew-field. The multiplicative group of a field is Abelian.



====Comments====
The finite multiplicative subgroups of skew-fields of finite non-zero characteristic are cyclic, and this is not the case in characteristic zero. There are only a finite number of even groups and an infinite number of odd groups, and the minimal order is 63. The classification is given in [[#References|[a1]]]. There exists a similar problem for proving a kind of Tits alternative: Any finite normal subgroup of the multiplicative group of a skew-field contains a free non-cyclic group or is a finitely-solvable group and has an extension to a linear group over a skew-field. Some cases are known, e.g., [[#References|[a2]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  S.A. Amitsur,   "Finite subgroups of division rings"  ''Trans. Amer. Math. Soc.'' , '''80'''  (1955)  pp. 361–396</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  A.I. Lichtman,   "Free subgroups in linear groups over some skew fields"  ''J. of Algebra'' , '''105'''  (1987)  pp. 1–28</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  W.R. Scott,   "Group theory" , Prentice-Hall  (1964)  pp. Chapt. 14, p. 426</TD></TR>
</table>