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Lie theory of finite simple groups and the Roth property

Research output: Contribution to journalArticle


Original languageEnglish
Pages (from-to)1-40
Number of pages40
Accepted/In press9 Jan 2017

King's Authors


In noncommutative geometry a ‘Lie algebra’ or bidirectional bicovariant differential calculus on a finite group is provided by a choice of an ad-stable generating subset (Formula presented.) stable under inversion. We study the associated Killing form K. For the universal calculus associated to (Formula presented.) = G \ {e} we show that the magnitude (Formula presented.) of the Killing form is defined for all finite groups (even when K is not invertible) and that a finite group is Roth, meaning its conjugation representation contains every irreducible, iff μ ≠ 1/(N − 1) where N is the number of conjugacy classes. We show further that the Killing form is invertible in the Roth case, and that the Killing form restricted to the (N − 1)-dimensional subspace of invariant vectors is invertible iff the finite group is an almost-Roth group (meaning its conjugation representation has at most one missing irreducible). It is known [9, 10] that most nonabelian finite simple groups are Roth and that all are almost Roth. At the other extreme from the universal calculus we prove that the 2-cycles conjugacy class in any Sn has invertible Killing form, and the same for the generating conjugacy classes in the case of the dihedral groups D 2n with n odd. We verify invertibility of the Killing forms of all real conjugacy classes in all nonabelian finite simple groups to order 75,000, by computer, and we conjecture this to extend to all nonabelian finite simple groups.

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