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7
votes
Accepted
Cyclic subgroups of finite abelian groups
I think that you can find the formulas that you are looking for in the paper "An arithmetic method of counting the subgroups of a finite abelian group" by Marius Tarnauceanu, Bull. Math. Soc. Sci. Mat …
3
votes
Results in the Presentation of Finite Groups
The result by Guralnick, Kantor, Kassabov and Lubotzky (which is mentioned already in Mark Sapir's answer) is one of the most striking results in finite group theory of the last decade, in my opinion. …