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Generating triples of involutions of large sporadic groups. (English. Russian original) Zbl 1057.20011

Discrete Math. Appl. 13, No. 3, 291-300 (2003); translation from Diskretn. Mat. 15, No. 2, 103-112 (2003).
In the Kourovka notebook [V. D. Mazurov and E. I. Khukhro (eds.), The Kourovka notebook. Unsolved problems in group theory, Novosibirsk (2002; Zbl 0999.20001)] the reviewer asked which finite simple nonabelian groups can be generated by three involutions two of which commute (Problem 7.30). At the present time, the answer is known [see V. D. Mazurov, Sib. Mat. Zh. 44, No. 1, 193-198 (2003); translation in Sib. Math. J. 44, No. 1, 160-164 (2003; Zbl 1035.20014)]). In particular, among the sporadic groups, only \(McL\), \(M_{11}\), \(M_{22}\) and \(M_{23}\) cannot be generated in this manner.
The author finds in every sporadic group, except \(F_1\), \(F_2\) and the above mentioned groups, an explicit triple of generating involutions two of which commute.

MSC:

20D08 Simple groups: sporadic groups
20F05 Generators, relations, and presentations of groups
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References:

[1] D. Gorenstein, Finite Simple Groups. An Introduction to Their Classification. Plenum Press, New York, 1982. · Zbl 0483.20008
[2] V. D. Mazurov and E. I. Khukhro, The Kourovka Notebook. Unsolved Problems in Group Theory. Inst. Math., Russian Acad. Sci., Siberian Div., Novosibirsk, 2002. · Zbl 0999.20001
[3] Ya N, Algebra Logic 36 pp 245– (1997)
[4] A. V. Timofeenko, On generating triples of some sporadic groups. Dep. VINITI 19.03.2001, No. 693-2001.
[5] A. V. Timofeenko, Generating triples of involutions of the Lyons and Janko groups J4. In: Abstr. Ukrainian Math. Congress. Inst. Math. Nat. Acad. Sci. Ukraine, Kiev, 2001, p. 50 (in Russian).
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