Finite groups with few dth roots of 1
Let G be a finite group. For each divisor d of \(| G|\), let \(\gamma_ d(G)=| \{x\in G| x^ d=1\}| /d\). By a well-known result of Frobenius, \(\gamma_ d(G)\) is a positive integer. Let \[ \gamma (G)=\max \{\gamma_ d(G)| \quad all\quad divisors\quad d\quad of\quad (G)\}. \] The authors call \(\gamma\) (G) the multiplicity of G. Let k be the maximum of the orders of elements of G and let \(i(G)=| G| /k\). The authors call i(G) the cocyclicity of G. This paper continues the study of the relationship between \(\gamma\) (G) and i(G). \textit{M. Hausman} and \textit{H. N. Shapiro} [Commun. Pure Appl. Math. 33, 635-649 (1980)] initiated the work by showing, inter alia, that there exists a function I(\(\gamma)\) such that for given \(\gamma =\gamma (G)\), i(G)\(\leq I(\gamma)\). Bounds on the size of I(\(\gamma)\) were obtained by the authors in an earlier paper [J. Algebra 91, 499-519 (1984)] and in this paper they obtain the (best possible) values for I(\(\gamma)\) for \(\gamma\leq 10\). In particular they show that \(I(10)=12.\) The paper also contains a number of interesting lemmas of a more general form. The arguments used involve Sylow theory and Zassenhaus' result on the structure of finite groups all of whose Sylow subgroups are cyclic.
scientific article; zbMATH DE number 3889788