Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T00:21:48.589Z Has data issue: false hasContentIssue false
  • English
  • Français

Enumeration of groups of prime-power order

Published online by Cambridge University Press:  17 April 2009

Brett E. Witty
Brett E. Witty
Affiliation:
9 Boronia Dr, Annandale, QLD 4814, Australia, e-mail: shorokin@hotmail com
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Finite group theorists have been interested in counting groups of prime-power order, as a preliminary step to counting groups of any finite order and to assist in explicitly listing such groups. In 1960, G. Higman considered when the functions giving the number of groups of prime-power order pn, for fixed n and varying p, is of a particular form, called polynomial on residue classes (PORC). The suggestion that such counting functions are PORC is known as Higman's PORC conjecture. In his 1960 paper [4] he proved that a certain class of groups of prime-power order, now called exponent-p class two groups, have counting functions that are PORC, but did not furnish explicit PORC functions.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 76 , Issue 3 , December 2007 , pp. 479 - 480
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Eick, B. and O'Brien, E.A., ‘Enumerating p-groups’, J. Austral. Math. Soc. Ser. A 67 (1999), 191–205.CrossRefGoogle Scholar
[2]Green, J.A., ‘The characters of the finite general linear groups,’ Trans. Amer. Math. Soc. 80 (1955), 402–447.CrossRefGoogle Scholar
[3]Higman, G., ‘Enumerating p-groups. I. Inequalities,’ Proc. London Math. Soc. (3) 10 (1960), 24–30.CrossRefGoogle Scholar
[4]Higman, G., ‘Enumerating p-groups. II. Problems whose solution is PORC’, Proc. London Math. Soc. (3) 10 (1960), 566–582.CrossRefGoogle Scholar
[5]Holt, D.F., Eick, B. and O'Brien, E.A., Handbook of computational group theory, Discrete Mathematics and its Applications (Chapman and Hall/CRC, Boca Raton, FL, 2005).CrossRefGoogle Scholar
[6]Steinberg, R., ‘A geometric approach to the representations of the full linear group over a Galois field,’ Trans. Amer. Math. Soc. 71 (1951), 274–282.CrossRefGoogle Scholar