Finite simple implies 2-generated

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., finite simple group) must also satisfy the second group property (i.e., 2-generated group)
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Definition

Any finite simple group is a 2-generated group: it has a generating set of size two. Note that since the simple abelian groups are cyclic of prime order, an equivalent formulation is that the minimum size of generating set of a finite simple non-abelian group is $2$.

Related facts

• Finite minimal simple implies 2-generated: This was proved by Thompson as a simple consequence of the classification of finite minimal simple groups, and proved later by Flavell without using a classification.
• Every finite group is generated by a solvable subgroup and one element
• Finite almost simple implies 3-generated
• Solvability is 2-local for finite groups

References

Journal references

• Some applications of the first cohomology group by Michael Aschbacher and Robert M. Guralnick, Journal of Algebra, ISSN 00218693, Volume 90,Number 2, Page 446 - 460(Year 1984): ^{Official copyMore info}
• 1-1/2-generation of simple groups by Alexander Stein, Beiträge Algebra Geom., Volume 39, Page 349 - 358(Year 1998): ^{More info}