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Subgroups of large finite groups and the linear optimization problem. (English. Russian original) Zbl 0631.20008
Algebra Logic 25, 254-260 (1986); translation from Algebra Logika 25, No. 4, 405-414 (1986).
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 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.
Reviewer: A.Kondrat’ev

##### MSC:
 20C15 Ordinary representations and characters 20D08 Simple groups: sporadic groups 20D05 Finite simple groups and their classification 20C10 Integral representations of finite groups 20-04 Software, source code, etc. for problems pertaining to group theory
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##### References:
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