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The MÖBius Function of Psl 2 ( q ), with Application to the Maximal Normal Subgroups of the Modular Group
Author(s) -
Downs Martin
Publication year - 1991
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-43.1.61
Subject(s) - modular group , group (periodic table) , mathematics , function (biology) , citation , library science , combinatorics , mathematics education , algebra over a field , computer science , physics , pure mathematics , biology , quantum mechanics , evolutionary biology
This paper seeks to prove some results stated in [5], namely a statement of the Mobius function of PSL2(2 ), e > 1. These results extend P. Hall's calculation in [6] of the Mobius function of PSL2(/?) for any prime/?. The techniques used in the present paper (see especially §2) should be of use for evaluating the Mobius function fiG for other classical groups G; in fact for the case when G = ?SL2(p ), where p is an odd prime and e > \,/iGis stated (without derivation) in §5. An application of our results is given by enumerating the normal subgroups N of the modular group T such that the quotient T/N is isomorphic to PSL2(/? ), for any prime p and e > 1. The idea of Mobius inversion on finite lattices is due to P. Hall in a rather neglected paper of 1936 [6]. It is very simple. If a is a real function on a finite lattice L (with ordering > and greatest element m) defined in terms of another real function 0 on L thus: