A090130 - OEIS

%I #20 Oct 15 2017 20:26:01

%S 1,1,2,5,15,77,684,34297

%N Number of groups of order 5^n.

%D G. Bagnera, La composizione dei Gruppi finiti il cui grado e la quinta potenza di un numero primo, Ann. Mat. Pura Appl. (3), 1 (1898), 137-228.

%D Hans Ulrich Besche, Bettina Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, International Journal of Algebra and Computation, Vol. 12, No 5 (2002), 623-644.

%D W. Burnside, Theory of Groups of Finite Order, Dover, NY, 1955.

%H M. F. Newman, E. A. O'Brien and M. R. Vaughan-Lee, <a href="http://dx.doi.org/10.1016/j.jalgebra.2003.11.012">Groups and nilpotent Lie rings whose order is the sixth power of a prime</a>, J. Algebra, 278 (2004), 383-401.

%H E. A. O'Brien and M. R. Vaughan-Lee, <a href="http://dx.doi.org/10.1016/j.jalgebra.2005.01.019">The groups of order p^7 for odd prime p</a>, J. Algebra 292, 243-258, 2005. [Eamonn O'Brien, Mar 06 2010]

%F For a prime p >= 5, the number of groups of order p^n begins 1, 1, 2, 5, 15, 61 + 2*p + 2*gcd (p - 1, 3) + gcd (p - 1, 4), 3*p^2 + 39*p + 344 + 24*gcd(p - 1, 3) + 11*gcd(p - 1, 4) + 2*gcd(p - 1, 5), ...

%o (GAP) A090130 := List([0..7],n -> NumberSmallGroups(5^n)); # _Muniru A Asiru_, Oct 15 2017

%Y Cf. A000001, A000679, A090091, A090140.

%K nonn

%O 0,3

%A Eamonn O'Brien (obrien(AT)math.auckland.ac.nz), Jan 22 2004

%E Corrected and extended by _David Radcliffe_, Feb 24 2010

%E Corrected and extended by Eamonn O'Brien, Mar 06 2010