Geodesic
Classification of indecomposable Abelian $(v,5)$-groups
Diskretnaya Matematika, Tome 20 (2008) no. 1, pp. 94-108

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The description of the $(v,3)$-groups was given previously, it appears that all such groups are Abelian. It is known that there are non-Abelian $(v,5)$-groups. In this paper, a description of the Abelian $(v,5)$-groups is given. 
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     title = {Classification of indecomposable {Abelian} $(v,5)$-groups},
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A. A. Frolov. Classification of indecomposable Abelian $(v,5)$-groups. Diskretnaya Matematika, Tome 20 (2008) no. 1, pp. 94-108. http://geodesic.mathdoc.fr/item/DM_2008_20_1_a8/