12

Such numbers have density $1$. In fact, for a density $1$ set of numbers it suffices to apply Sylow's theorem to the largest prime factor. In other words, for a density $1$ set of numbers $n$, with $p$ the largest prime factor of $n$, there is no number congruent to $1$ mod $p$ dividing $n$ other than $1$. Proof: We start from a classical fact of analytic ...


2

First get the possible types: gap> o:=Set(List(ConjugacyClasses(G),x->Order(Representative(x)))); [ 1, 2, 3, 4, 7 ] gap> t:=Filtered(UnorderedTuples(o,3),x->1/x[1]+1/x[2]+1/x[3]<1); [ [ 2, 3, 7 ], [ 2, 4, 7 ], [ 2, 7, 7 ], [ 3, 3, 4 ], [ 3, 3, 7 ], [ 3, 4, 4 ], [ 3, 4, 7 ], [ 3, 7, 7 ], [ 4, 4, 4 ], [ 4, 4, 7 ], [ 4, 7, 7 ], [ 7, 7, 7 ] ] ...


1

Since it is a bit too long for a comment: In the development version of GAP, my laptop computed the result for $A_{32}$ in about 20 minutes and 1.5GB: rec( inclusions := [ [ 0, 1 ], [ 0, 2 ], [ 0, 3 ], [ 0, 6 ], [ 1, 4 ], [ 1, 5 ], [ 1, 7 ], [ 2, 5 ], [ 2, 8 ], [ 2, 9 ], [ 3, 4 ], [ 3, 9 ], [ 3, 12 ], [ 4, 11 ], [ 4, 13 ], [ 5, 10 ], [ 5, 11 ], [ ...


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