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# Structures in Units of Group Rings and Twisted Group Rings (FWOTM880)

Denote by Z the integers and let G be a finite multiplicative group. View G as a basis of a "vector space" over Z - this defines the set of elements in the integral group ring ZG, i.e. every element in ZG is a linear combination of elements in G with integral coefficients. As in vector spaces one can define an additive structure on ZG by adding coefficients corresponding to the same basis element. E.g. if g, h are elements in G, then (3g - 2h) + (-4g+4h) = -g + 2h.

Now the group structure of G also gives rise to a multiplicative structure * on ZG by assuming that an integer commutes with any element from G. Assume e.g. that the elements g and h in G commute, i.e. we have gh = hg, then (3g - 2h)*(-4g+4h) = -12g^2 + 12gh + 8hg - 8h^2 = -12g^2 + 20gh - 8h^2. As the group G contains a neutral element e for multiplication and the integers the neutral element 1, we obtain also a neutral element for multiplication in ZG, namely 1*e.

The object of study of this proposal is the unit group of ZG, i.e. the elements in ZG which can be multiplied with an other element in ZG to obtain the neutral element 1*e. More precisely, we will investigate the connection between the structure of the unit group and the group base G and how much of the group theoretical properties of G are reflected in the structure of the unit group. A central question is a conjecture made by H. J. Zassenhaus in the 1970's stating that all units of finite order are basically given by the elements in G and -G.
Date:1 Oct 2017 →  30 Sep 2020
Keywords:group rings, structure
Disciplines:General mathematics