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On Graham Higman's famous PORC paper | ||
| International Journal of Group Theory | ||
| مقاله 6، دوره 1، شماره 4، اسفند 2012، صفحه 65-79 اصل مقاله (448.12 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2012.1591 | ||
| نویسنده | ||
| Michael Vaughan-Lee* | ||
| Oxford University Mathematical Institute | ||
| چکیده | ||
| We investigate Graham Higman's paper Enumerating $p$-groups, II, in which he formulated his famous PORC conjecture. We are able to simplify some of the theory. In particular, Higman's paper contains five pages of homological algebra which he uses in his proof that the number of solutions in a finite field to a finite set of monomial equations is PORC. It turns out that the homological algebra is just razzle dazzle, and can all be replaced by the single observation that if you write the equations as the rows of a matrix then the number of solutions is the product of the elementary divisors in the Smith normal form of the matrix. We obtain the PORC formulae for the number of $r$-generator groups of $p$ -class two for $r\leq 6$. In addition, we obtain the PORC formula for the number of $p$-class two groups of order $p^{8}$. | ||
| کلیدواژهها | ||
| Enumerating p-groups؛ PORC conjecture؛ Graham Higman | ||
| مراجع | ||
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