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A090140 Number of groups of order 7^n. 7

%I #11 Apr 07 2013 08:09:09

%S 1,1,2,5,15,83,860,113147

%N Number of groups of order 7^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.

%D 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.

%D 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.

%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), ...

%Y Cf. A000001, A000679, A090091, A090130.

%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 Updated reference for p^7 Eamonn O'Brien (obrien(AT)math.auckland.ac.nz), Mar 06 2010

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