Geodesic
CLASSIFICATION OF FINITE GROUPS VIA THEIR BREADTH
Glasgow mathematical journal, Tome 62 (2020) no. 3, pp. 544-563

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DOI

Let k be a divisor of a finite group G and Lk(G) = {x ∈ G | xk =1}. Frobenius proved that the number |Lk(G)| is always divisible by k. The following inverse problem is considered: for a given integer n, find all groups G such that max{k-1|Lk(G)| | k ∈ Div(G)} = n, where Div(G) denotes the set of all divisors of |G|. A procedure beginning with (in a sense) minimal members and deducing the remaining ones is outlined and executed for n=8. 
HEINEKEN, HERMANN; RUSSO, FRANCESCO G. CLASSIFICATION OF FINITE GROUPS VIA THEIR BREADTH. Glasgow mathematical journal, Tome 62 (2020) no. 3, pp. 544-563. doi: 10.1017/S0017089519000272
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     author = {HEINEKEN, HERMANN and RUSSO, FRANCESCO G.},
     title = {CLASSIFICATION {OF} {FINITE} {GROUPS} {VIA} {THEIR} {BREADTH}},
     journal = {Glasgow mathematical journal},
     pages = {544--563},
     year = {2020},
     volume = {62},
     number = {3},
     doi = {10.1017/S0017089519000272},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089519000272/}
}
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