Groups of order $p^8$ and exponent $p$

Document Type: Research Paper

Author

Oxford University Mathematical Institute

Abstract

‎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.

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