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Some definitions of Klein's simple group of order 168 and other groups
Published online by Cambridge University Press: 18 May 2009
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In a previous note [3], Mennicke and I showed that the relations
(8, 7|2, 3): A8=B7=(AB)2=(AB)2=(A-1B)2=(A-1B)3=E
define a group of order 10752. As we remarked, the results of §§ 2, 3 of that note are not restricted in their application to this group; they apply to the group
[3, 7]+: B7=(AB)2=(A-1B)3=E
and to any factor group of this group which in turn has Klein's simple group of order 168, defined by
(4,7|2, 3): A4=B7=(AB)2=(A-1B)3=E,
as a factor group. In this note I use these results to establish alternative “weaker” definitions for Klein's group and for two groups discussed by Sinkov [4], namely (8, 7|2, 3) defined above and a factor group of this group of order 1344. These latter groups are eloquently discussed by Coxeter [1].
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- Copyright © Glasgow Mathematical Journal Trust 1962
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