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Abstract Root Subgroups and Simple Groups of Lie-Type (Hardcover, 2001 ed.)
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Abstract Root Subgroups and Simple Groups of Lie-Type (Hardcover, 2001 ed.)
Series: Monographs in Mathematics, 95
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It was already in 1964 Fis66] when B. Fischer raised the question:
Which finite groups can be generated by a conjugacy class D of
involutions, the product of any two of which has order 1, 2 or 37
Such a class D he called a class of 3-tmnspositions of G. This
question is quite natural, since the class of transpositions of a
symmetric group possesses this property. Namely the order of the
product (ij)(kl) is 1, 2 or 3 according as {i, j} n {k, l} consists
of 2,0 or 1 element. In fact, if I{i, j} n {k, I}1 = 1 and j = k,
then (ij)(kl) is the 3-cycle (ijl). After the preliminary papers
Fis66] and Fis64] he succeeded in Fis71J, Fis69] to classify all
finite "nearly" simple groups generated by such a class of
3-transpositions, thereby discovering three new finite simple
groups called M(22), M(23) and M(24). But even more important than
his classification theorem was the fact that he originated a new
method in the study of finite groups, which is called "internal
geometric analysis" by D. Gorenstein in his book: Finite Simple
Groups, an Introduction to their Classification. In fact D.
Gorenstein writes that this method can be regarded as second in
importance for the classification of finite simple groups only to
the local group-theoretic analysis created by J. Thomp
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