Multiplicative group

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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.


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].


[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:
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article