Groups of order p8 and exponent p

Document Type : Research Paper


Oxford University Mathematical Institute


‎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.


Main Subjects

G. Bagnera (1898). La composizione dei gruppi finiti il cui grado e la quinta potenza di un numero primo. Ann. Mat. Pura Appl.. 3 (1), 137-228 H. R. Brahana (1951). Finite metabelian groups and the lines of a projective four-space. Amer. J. Math.. 73, 539-555 J. Cannon, W. Bosma and C. Playoust (1997). The Magma algebra system I: The user language. J. Symbolic Comput.. 24, 235-265 F. N. Cole and J. W. Glover (1893). On groups whose orders are products of three prime factors. Amer. J. Math.. 15, 191-220 A. Cop etti (2005). Finite-dimensional Lie algebras of nilpotency class 2. Masters Thesis, Australian National University. M. du Sautoy and M. Vaughan-Lee (2012). Non-PORC behaviour of a class of descendant p-groups. J. Algebra. 361, 287-312 B. Eick, H. U. Besche and E. A. O'Brien (2001). The groups of order at most 2000. Electron. Res. Announc. Amer. Math. Soc.. 7, 1-4 A. Evseev (2008). Higman's PORC conjecture for a family of groups. Bul l. Lond. Math. Soc.. 40, 415-431 The GAP Group (2014). GAP-Groups, Algorithms, and Programming, Version 4.7.5. G. Higman (1960). Enumerating p-groups. I: Inequalities. Proc. London Math. Soc. (3). 10, 24-30 G. Higman (1960). Enumerating p -groups. II: Problems whose solution is PORC. Proc. London Math. Soc. (3). 10, 566-582 O. Holder (1893). Die Grupp en der Ordnungen p^3, pq^2, pq^r, p^4. Math. Ann.. 43, 301-412 E. Netto (1882). Substitutionentheorie und ihre Anwendungen auf die Algebra. Teubner, Leipzig. M. F. Newman, E. A. O'Brien and M. R. Vaughan-Lee (2004). Groups and nilpotent Lie rings whose order is the sixth power of a prime. J. Algebra. 278, 383-401 E. A. O'Brien (1990). The p-group generation algorithm. J. Symbolic Comput.. 9, 677-698 E. A. O'Brien (1991). The groups of order 256. J. Algebra. 143, 219-235 E. A. O'Brien and M. R. Vaughan-Lee (2005). The groups with order p^7 for odd prime p. J. Algebra. 292, 243-258 M. Vaughan-Lee (2012). On Graham Higman's famous PORC paper. Int. J. Group Theory. 1 (4), 65-79 M. Vaughan-Lee and B. Eick (2013). LiePRing - a GAP package. Version 1.5, Packages/liepring.html. J. W. A. Young (1893). On the determination of groups whose order is a power of a prime. Amer. J. Math.. 15, 124-178