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\(3/2\)-generation of the sporadic simple groups. (English) Zbl 0806.20020
A group \(G\) is said to be \(3/2\)-generated if, given an arbitrary non-identity element \(x\) of \(G\) one can always find an element \(y=y(x)\) of \(G\) such that \(\langle x,y\rangle=G\). Such an element \(y\) is called complementary. L. Di Martino and M. C. Tamburini conjectured [in Generators and relations in groups and geometries, NATO ASI Ser., Ser. C 333, 195-233 (1991; Zbl 0751.20027)] that every finite simple group is \(3/2\)-generated. In this note the author verifies the \(3/2\)-generation conjecture for the family of sporadic simple groups. In fact, he proves that if \(G\) is a sporadic simple group and \(p\) is the maximum prime divisor of \(G\) then, given arbitrary \(x\neq 1\) in \(G\), a complementary \(y\) can always be chosen from the conjugacy class \(pA\) of \(G\) with notation as in the “Atlas of Finite Groups” [Oxford (1985; Zbl 0568.20001)].

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