Power-commutator presentation

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Definition

A power-commutator presentation of a group G for a prime number p is a presentation with generating set a_i, i \in I for a totally ordered indexing set I and relations of the form:

  • power relations: a_i^p is written as a product of powers of a_k, k > i, with the ks in increasing order as we go from left to right in the product. The exponent on a_k for k > i is denoted \beta(i,k).
  • commutator relations: The commutator [a_i,a_j] is written as a product of powers of a_k, k > \max \{ i , j \}, with the ks in increasing order as we go from left to right in the product. The exponent of a_k for k > \max \{ i , j \} is denoted \beta(i,j,k).

For a group of prime power order p^n, a power-commutator presentation is termed consistent if it uses exactly n generators.

Facts