Descriptions of Groups of Primepower Order

It is well known, and was first proved by Sylow (1872), that every group of prime-power order p has a presentation of the form {a1, . . . , an | api = vii, 1 ≤ i ≤ n, [ak, aj ] = vjk, 1 ≤ j < k ≤ n}, where vjk is a word in the elements ak+1, . . . , an for 1 ≤ j ≤ k ≤ n . A presentation of this form is now called a power-commutator presentation for the group and written briefly {A | R} . As was first observed by Neubüser (1961), these presentations provide a good basis for working with groups of prime-power order using computers. The recent text of Sims (1994) provides much general background; we will refer to it with page numbers. In terms of a generating set of this kind every element of the group can be written uniquely as a normal word, which is a word of the form a 1 · · · a ξ(n) n with 0 ≤ ξ(i) < p . Sims (1994, p. 395) calls the above words collected. We can assume without loss of generality that the words vjk occurring on the right-hand sides of the relations in a power-commutator presentation are normal. The fundamental importance of a power-commutator presentation {A | R} is that given a word in A the relations in R can be used to compute mechanically a normal word representing the same element of the group and hence to compute products and inverses. Power-commutator presentations play a similar core role in the theory of groups of prime-power order to that played by bases and strong generating sets for permutation groups (e.g. Butler, 1991). They provide a comfortable and uniform way of describing the groups in question. This often makes it possible to divide the task of studying groups of prime-power order into three steps—computing a power-commutator presentation from