The groups of exponent p and order \(p^ 7\) (p any prime) (Q1106321)

This paper aims to give a list of groups of exponent p and order \(p^ 7\) for all primes p. The list consists of commutator presentations on 7 generators. It is intended that each group of exponent p and order \(p^ 7\) be isomorphic to exactly one group in the list. This intention is not achieved. There is (at least one) duplication: group 117 is isomorphic to group 115 when 4 divides p-1 and to group 116 when 4 divides p-3. There are at least two groups which do not occur in the list. (I am indebted to D. B. Tyler for drawing my attention to these errors and E. A. O'Brien for helping to check them.)