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A cyclic number is a natural number $n$ such that any group of order $n$ is cyclic. A003277

Theorem (T. Szele, 1947): A number $n$ is cyclic if and only if it is coprime to its Euler totient $\varphi(n)$. Proof: see here.

"As usual, a finite-dimensional Hopf algebra is called trivial if it or its dual is a group algebra."

Question: Is a semisimple Hopf algebra of cyclic dimension trivial?

By the theorem, a number $n$ cyclic iff it is square-free $\prod_{i=1}^r p_i$ and $\forall i, j$, $p_i$ is coprime to $p_j-1$.
Now, a semisimple Hopf algebra of dimension $p_1$ or $p_1p_2$ is trivial, so we need to deal with $r\ge 3$.

The first cyclic numbers with $r=3$ are $3^1 5^1 17$, $3^1 5^1 23$, $3^1 5^1 29$, $5^1 7^1 13$, $3^1 11^1 17$.

gap> L:=[];; P:=Filtered([1..101], IsPrime);; for p in P do for q in P do for r in P do if p<q and q<r and GcdInt(p,q-1)=1 and GcdInt(p,r-1)=1 and  GcdInt(q,r-1)=1 and p*q*r <= 3*5*101 then Add(L,p*q*r); fi; od; od; od; Sort(L); L;
[ 255, 345, 435, 455, 561, 595, 665, 705, 795, 805, 885, 957, 1001, 1105, 1173, 1235, 1245, 1295, 1309, 1335, 1353, 1463, 1479, 1495 ]

An integral fusion category (or semisimple Hopf algebra) of dimension $p_1p_2p_3$ is group-theoretical.

Bonus question 1: Can we deduce a positive answer of the main question for $p_1p_2p_3$?

Bonus question 2: Is an integral fusion category of cyclic dimension weakly group-theoretical?
(A weakly group-theoretical integral fusion category of square-free dimension is group-theoretical.)

Bonus question 3: If so, can we answer the main question in general?

The first cyclic numbers with $r=4$ are $3^1 5^1 17^1 23$, $3^1 5^1 17^1 29$, $5^1 7^1 13^1 17$, $ 5^1 7^1 13^1 19$.

gap> L:=[];; P:=Filtered([1..211], IsPrime);; for p in P do for q in P do for r in P do for s in P do if p<q and q<r and r<s and GcdInt(p,q-1)=1 and GcdInt(p,r-1)=1 and GcdInt(p,s-1)=1 and GcdInt(q,r-1)=1 and GcdInt(q,s-1)=1 and GcdInt(r,s-1)=1 and p*q*r*s <= 3*5*7*211 then Add(L,p*q*r*s); fi; od; od; od; od; Sort(L); L;  
[ 5865, 7395, 7735, 8645, 10005, 10465, 11305, 11985, 13515, 13685, 15045, 15295, 16269, 16835, 17017, 18285, 19019, 20355, 20445, 20995, 21165, 21385, 22015 ]

An abelian number is a natural number $n$ such that any group of order $n$ is abelian. A051532
Theorem (Dickson): A number $\prod_i p_i^{n_i}$ is abelian iff $\forall i, j$, $n_i\le 2$ and $p_i$ is coprime to $p_j^{n_j}-1$.
Proof: See here.

A nilpotent number is a natural number $n$ such that any group of order $n$ is nilpotent. A056867
Theorem (Schmidt): A number $\prod_i p_i^{n_i}$ is nilpotent iff $\forall i, j$, $p_i$ is coprime to $p_j^{\ell}-1$ with $1 \le \ell \le n_j$. Proof: See here.

We could extend Bonus question 2 to an abelian dimension, and then to a nilpotent dimension.

Warning: A weakly group-theoretical fusion category is (by definition) a fusion category which is Morita equivalent to a nilpotent fusion category, but the representation category of any finite group is a nilpotent fusion category, this notion just extends categorically the notion of a finite nilpotent group.

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  • 3
    $\begingroup$ What do you mean by "trivial"? A group Hopf algebra? $\endgroup$ – darij grinberg Jul 12 '17 at 16:07
  • $\begingroup$ Also, this paper might be of help -- if your concept of "trivial" deserves this name, then I'd say these Hopf algebras would have to have a simpler structure than semisolvability. See also the "two classification results" cited on p. 104. $\endgroup$ – darij grinberg Jul 12 '17 at 16:10
  • $\begingroup$ @darijgrinberg: "As usual, a finite-dimensional Hopf algebra is called trivial if it or its dual is a group algebra". I don't understand your second comment because the paper you cite is about dimension $2q^3$ whereas a cyclic number is square-free. $\endgroup$ – Sebastien Palcoux Jul 12 '17 at 18:04
  • $\begingroup$ Ouch! I misread that definition as "the group of units of $\mathbb {Z} / n$ is cyclic". $\endgroup$ – darij grinberg Jul 12 '17 at 18:11

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