A090130 Number of groups of order 5^n.

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

Offset

0,3

References

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.

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.

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

Links

Table of n, a(n) for n=0..7.

M. F. Newman, E. A. O'Brien and M. R. Vaughan-Lee, Groups and nilpotent Lie rings whose order is the sixth power of a prime, J. Algebra, 278 (2004), 383-401.

E. A. O'Brien and M. R. Vaughan-Lee, The groups of order p^7 for odd prime p, J. Algebra 292, 243-258, 2005. [Eamonn O'Brien, Mar 06 2010]

Formula

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), ...

Prog

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

Crossrefs

Cf. A000001, A000679, A090091, A090140.

Sequence in context: A090091 A268122 A133645 * A090140 A032267 A191477

Adjacent sequences: A090127 A090128 A090129 * A090131 A090132 A090133

Keyword

nonn

Author

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

Extensions

Corrected and extended by David Radcliffe, Feb 24 2010

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

Status

approved