A051532 - OEIS

A051532

The abelian orders (or abelian numbers): numbers m such that every group of order m is abelian.

1, 2, 3, 4, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 29, 31, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 99, 101, 103, 107, 109, 113, 115, 119, 121, 123, 127, 131, 133, 137, 139, 141, 143, 145, 149, 151, 153, 157, 159, 161

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OFFSET

1,2

COMMENTS

Except for a(2)=2 and a(4)=4, all of the terms in the sequence are odd. This is because of the existence of a non-abelian dihedral group of order 2m for each m > 2.

Cubefree terms of A056867; A212793(a(n)) = 1. - Reinhard Zumkeller, Jun 28 2013

See similar comment with "squarefree terms" in A003277 (Donald J. McCarthy link). - Bernard Schott, Feb 20 2023

REFERENCES

W. R. Scott, Group Theory, Dover, 1987, page 217.

LINKS

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Donald J. McCarthy, A survey of partial converses to Lagrange's theorem on finite groups, Transactions of the New York Academy of Sciences, 1971, page 592.

J. Pakianathan and K. Shankar, Nilpotent Numbers, Amer. Math. Monthly, 107, August-September 2000, pp. 631-634.

Index entries for sequences related to groups

FORMULA

m must be cubefree and its prime divisors must satisfy certain congruences.

Let the prime factorization of m be p1^e1 * ... * pr^er. Then m is in this sequence if ei < 3 for all i and pi^k is not congruent to 1 (mod pj) for all i and j and 1 <= k <= ei. - T. D. Noe, Mar 25 2007

EXAMPLE

a(4) = 4 because every group of order 4 is abelian.

These two abelian groups of order 4 are the cyclic group C_4 and the Klein four-group = C_2 X C_2, the smallest non-cyclic abelian group. - Bernard Schott, Feb 21 2023

MATHEMATICA

okQ[n_] := Module[{f, lf, p, e, v}, f = FactorInteger[n]; lf = Length[f]; p = f[[All, 1]]; e = f[[All, 2]]; If[AnyTrue[e, # > 2&], Return[False]]; v = p^e; For[i = 1, i <= lf, i++, For[j = i+1, j <= lf, j++, If[Mod[v[[i]], p[[j]]]==1 || Mod[v[[j]], p[[i]]]==1, Return[False]]]]; Return[True]];

Select[Range[200], okQ] (* Jean-François Alcover, May 03 2012, after PARI, updated Jan 10 2020 *)

PROG

(PARI) is(n)=my(f=factor(n), v=vector(#f[, 1])); for(i=1, #v, if(f[i, 2]>2, return(0), v[i]=f[i, 1]^f[i, 2])); for(i=1, #v, for(j=i+1, #v, if(v[i]%f[j, 1]==1 || v[j]%f[i, 1]==1, return(0)))); 1 \\ Charles R Greathouse IV, Feb 13 2011

(Haskell)

a051532 n = a051532_list !! (n-1)

a051532_list = filter ((== 1) . a212793) a056867_list

-- Reinhard Zumkeller, Jun 28 2013

(Python)

from sympy import factorint

def ok(n):

if n == 1: return True

f = factorint(n)

p, e = f.keys(), f.values()

if max(e) >= 3: return False

return all((pi**k)%pj!=1 for pi in p for pj in p if pj!=pi for k in range(1, f[pi]+1))