## Groups of order $$p^8$$ and exponent $$p$$.(English)Zbl 1456.20016

Summary: We prove that for $$p>7$$ there are $p^4+2p^3+20p^2+147p+(3p+29)\gcd(p-1,3)+5 \gcd(p-1,4)+1246$ groups of order $$p^8$$ with exponent $$p$$. If $$P$$ is a group of order $$p^8$$ and exponent $$p$$, and if $$P$$ has class $$c>1$$ then $$P$$ is a descendant of $$P/\gamma_c(P)$$. For each group of exponent $$p$$ with order less than $$p^8$$ we calculate the number of descendants of order $$p^8$$ with exponent $$p$$. In all but one case we are able to obtain a complete and irredundant list of the descendants. But in the case of the three generator class two group of order $$p^6$$ and exponent $$p (p>3)$$, while we are able to calculate the number of descendants of order $$p^8$$, we have not been able to obtain a list of the descendants.

### MSC:

 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20D15 Finite nilpotent groups, $$p$$-groups

### Keywords:

PORC; $$p$$-groups; enumeratio

### Software:

GAP; Magma; LiePRing
Full Text:

### References:

 [1] G. Bagnera, La composizione dei gruppi finiti il cui grado ‘e la quinta potenza di un numero primo,Ann. Mat. Pura Appl.,3no.1 (1898) 137-228. · JFM 29.0112.03 [2] H. R. Brahana, Finite metabelian groups and the lines of a projective four-space,Amer. J. Math.,73(1951) 539-555. · Zbl 0043.02802 [3] J. Cannon, W. Bosma and C. Playoust, The Magma algebra system I: The user language,J. Symbolic Comput.,24 (1997) 235-265. · Zbl 0898.68039 [4] F. N. Cole and J. W. Glover, On groups whose orders are products of three prime factors,Amer. J. Math.,15 (1893) 191-220. · JFM 25.0205.02 [5] A. Copetti,Finite-dimensional Lie algebras of nilpotency class 2, Masters Thesis, Australian National University, 2005. [6] M. du Sautoy and M. Vaughan-Lee, Non-PORC behaviour of a class of descendantp-groups,J. Algebra,361(2012) 287-312. · Zbl 1267.20025 [7] B. Eick, H. U. Besche and E. A. O’Brien, The groups of order at most 2000,Electron. Res. Announc. Amer. Math. Soc.,7(2001) 1-4. · Zbl 0986.20024 [8] A. Evseev, Higman’s PORC conjecture for a family of groups,Bull. Lond. Math. Soc.,40(2008) 415-431. · Zbl 1147.20014 [9] The GAP Group,GAP - Groups, Algorithms, and Programming, Version 4.7.5, 2014,http://www.gap-system.org. [10] G. Higman, Enumeratingp-groups. I: Inequalities,Proc. London Math. Soc. (3),10(1960) 24-30. · Zbl 0093.02603 [11] G. Higman, Enumeratingp-groups. II: Problems whose solution is PORC,Proc. London Math. Soc. (3),10(1960) 566-582. · Zbl 0201.36502 [12] O. H¨older, Die Gruppen der Ordnungenp3,pq2,pqr,p4,Math. Ann.,43(1893) 301-412. · JFM 25.0201.02 [13] E. Netto,Substitutionentheorie und ihre Anwendungen auf die Algebra, Teubner, Leipzig (1882). · JFM 14.0090.01 [14] 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. · Zbl 1072.20022 [15] E. A. O’Brien, Thep-group generation algorithm,J. Symbolic Comput.,9(1990) 677-698. · Zbl 0736.20001 [16] E. A. O’Brien, The groups of order 256,J. Algebra,143(1991) 219-235. · Zbl 0734.20001 [17] E. A. O’Brien and M. R. Vaughan-Lee, The groups with orderp7for odd primep,J. Algebra,292(2005) 243-258. · Zbl 1108.20016 [18] M. Vaughan-Lee, On Graham Higman’s famous PORC paper,Int. J. Group Theory,1no. 4 (2012) 65-79. · Zbl 1267.20024 [19] M. Vaughan-Lee and B. Eick,LiePRing – a GAP package, Version 1.5, (2013),http://www.gap-system.org/ Packages/liepring.html. [20] J. W. A. Young, On the determination of groups whose order is a power of a prime,Amer. J. Math.,15(1893) 124-178. · JFM 25.0201.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.