Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Lie theory of finite simple groups and the Roth property   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 C stable under inversion. We study the associated Killing
form. For the universal calculus associated to C=G \ {e} we show that the
magnitude of the Killing form \mu=\sum_{a,b\in C}K^{-1}_{a,b} 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
\mu\ is not equal to 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 almost-Roth group (meaning its
conjugation representation has at most one missing irreducible). It is known
that most finite simple groups are Roth and that all are almost Roth. At the
other extreme from the universal calculus we prove that the generating
conjugacy class in the case of the dihedral groups D_{2n} with n odd has
invertible Killing form, and the same for the 2-cycles conjugacy class in any
S_n. We also compute some eigenvalues of the Killing form in the case of the
n-cycles class in S_n. Finally, 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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The categories of this paper are: patt-sol
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The categories of this paper are: cond-mat
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-th
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The categories of this paper are: comp-gas
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The categories of this paper are: nlin
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: funct-an
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The categories of this paper are: eess
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The categories of this paper are: chem-ph
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The categories of this paper are: cs
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The categories of this paper are: math-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: stat
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The categories of this paper are: nucl-th