Subgroups of large finite groups and the linear optimization problem
Let G be a finite group and let H be a subgroup of G with \(| G:H| \leq m\). Assume that the character table of G is known. Let \(\psi_ i\) denote the sum \(\phi_{i_ 1}+\phi_{i_ 2}+...+\phi_{i_ t}\) for the i-th algebraical conjugacy class \(\{\phi_{i_ 1},\phi_{_ 2},...,\phi_{i_ t}\}\) of complex irreducible characters of G. A linear combination \(\sum x_ i\psi_ i\) is said to be a pseudo- permutation character of degree s of G if the following conditions (1) [see the author and \textit{V. D. Mazurov}, Mat. Zametki 37, No.2, 145-151 (1985; Zbl 0579.20012)] and (2) hold: \(\sum x_ i\psi_ i(1)\leq m\); \(\sum x_ i\psi_ i(g)\geq 0\) for every \(g\in G\); \(\sum x_ i(\psi_ i(g^ k)-\psi_ i(g))\geq 0\) for every \(g\in G\) and every \(k\in {\mathbb{N}}\); \(x_ 1=1\) for the principal character \(\psi_ 1=1_ G\); \(x_ i\geq 0\); and (2) \(x_ i\in {\mathbb{Z}}\), \(s=\sum x_ i\psi_ i(1)\) is the divisor of \(| G|.\) The permutation character of G on the cosets of H is a pseudo- permutation. The existence of a pseudo-permutation character of degree m does not guarantee the existence of a subgroup of index m, but is a strong necessary condition for such existence. In this paper the author states a method of determination of pseudo-permutation characters with the help of a computer. This method is based, in particular, on the simplex-method applied in linear programming. As an illustration it is computed the characters of the permutation representations of simple sporadic groups \(F_ 2\), \(F_ 3\), \(F_ 5\) and \(J_ 4\) on the cosets of proper subgroups of minimal index.
scientific article; zbMATH DE number 4025673