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A319597
Number of conjugacy classes for a non-abelian group of order p^3, where p is prime: a(n) = p^2 + p - 1 where p = prime(n).
1
5, 11, 29, 55, 131, 181, 305, 379, 551, 869, 991, 1405, 1721, 1891, 2255, 2861, 3539, 3781, 4555, 5111, 5401, 6319, 6971, 8009, 9505, 10301, 10711, 11555, 11989, 12881, 16255, 17291, 18905, 19459, 22349, 22951, 24805, 26731, 28055, 30101, 32219, 32941, 36671
OFFSET
1,1
COMMENTS
For a non-abelian group of order p^3, we can use the class equation, p-group has nontrivial center result, group modulo center is cyclic implies group is abelian result, and the orbit-stabilizer theorem to give the number of conjugacy classes and number of elements in each conjugacy class.
The elements of A028387 with prime index.
LINKS
FORMULA
From Amiram Eldar, Nov 07 2022: (Start)
a(n) = A028387(A000040(n)-1).
Product_{n>=1} (1 + 1/a(n)) = A065489.
Product_{n>=1} (1 - 1/a(n)) = A065480. (End)
EXAMPLE
For p^3=2^3=8, the conjugacy classes of the Dihedral group = <r, s | r^4=1, s^2=1, srs=r^{-1}> are {1}, {r^2}, {r, r^3}, {s, sr^2}, {sr, sr^3}.
MAPLE
A028387:= n -> n^2+n-1:
seq(A028387(ithprime(i)), i=1..50); # Robert Israel, Dec 23 2018
MATHEMATICA
f[n_]:=n^2 + n - 1 ; f[Prime[Range[43]]] (* Amiram Eldar, Nov 21 2018 *)
PROG
(PARI) a(n) = {my(p = prime(n)); p^2 + p - 1; } \\ Amiram Eldar, Nov 07 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Juan Lanfranco, Sep 23 2018
STATUS
approved