Namespaces
Variants
Actions

Multiplicative group

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

2020 Mathematics Subject Classification: Primary: 12E15 [MSN][ZBL]

of a skew-field

The group of all elements of the given 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 [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., [a2].

References

[a1] S.A. Amitsur, "Finite subgroups of division rings" Trans. Amer. Math. Soc. , 80 (1955) pp. 361–396
[a2] A.I. Lichtman, "Free subgroups in linear groups over some skew fields" J. of Algebra , 105 (1987) pp. 1–28
[a3] W.R. Scott, "Group theory" , Prentice-Hall (1964) pp. Chapt. 14, p. 426
How to Cite This Entry:
Multiplicative group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multiplicative_group&oldid=35792
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article