Celebratio Mathematica — Albert

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Works connected to John Griggs Thompson

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A. A. Albert and J. Thompson: “Two element generation of the projective unimodular group,” Bull. Am. Math. Soc. 64 (1958), pp. 92–93. MR 0096728 Zbl 0082.02605 article

A. A. Albert and J. Thompson: “Two-element generation of the projective unimodular group,” Ill. J. Math. 3 : 3 (1959), pp. 421–439. MR 0106951 Zbl 0098.02302 article

Let \( \mathfrak{F} = \mathfrak{F}_q \) be the field of \( q = p^m \) elements, \( \mathfrak{M} = \mathfrak{M}(n,q) \) the multiplicative group of all \( n \)-rowed square matrices with elements in and determinant 1, and \( \mathfrak{N} = \mathfrak{N}(n,q) \) the subgroup of \( \mathfrak{M} \) consisting of its scalar matrices \( \rho I \) with \( \rho^n = 1 \). We assume, of course, that \( n > 1 \). Then \( \mathfrak{N} \) is a normal subgroup of \( \mathfrak{N} \), and the quotient group \[ \mathfrak{G}=\mathfrak{G}(n,q)=\mathfrak{M}/\mathfrak{N} \] is a well-known simple group called the projective unimodular group.

In [1930] H. R. Brahana gave a list of simple groups of orders less than 1,000,000. An examination of his list reveals the fact that every group there is generated by two elements, one of which has period (group order) two. The purpose of this paper is to prove the corresponding result for a general class of simple groups. We shall derive the following property.

The projective unimodular group is generated by two elements \( A\mathfrak{N} \) and \( B\mathfrak{N} \), where the coset \( A\mathfrak{N} \) has period two.