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

Vaughan-Lee, Michael

Int. J. Group Theory 4, No. 4, 25-42 (2015).

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

References:

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