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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)].

##### MSC:
 20D08 Simple groups: sporadic groups 20F05 Generators, relations, and presentations of groups
##### Keywords:
$$3/2$$-generations; sporadic simple groups
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##### References:
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