Power-commutator presentation: Difference between revisions

Revision as of 03:48, 1 November 2015

Definition

A power-commutator presentation of a group $G$ for a prime number $p < /, a t h > i s a [[p r e s e n t a t i o n]] w i t h g e n e r a t i n g s e t < m a t h > a_{i}, i \in I$ 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 $k$ s in increasing order as we go from left to right in the product. The exponent on $a_{k}$ for $k > i$ is denoted $β (i, k)$ .
commutator relations: The commutator $[a_{i}, a_{j}]$ is written as a product of powers of $a_{k}, k > max {i, j}$ , with the $k$ s 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 $β (i, j, k)$ .

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

Facts

Every group of prime power order has a power-commutator presentation

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 ==Definition==
-A '''power-commutator presentation''' of a [[group]] <math>G</math> is a [[presentation]] with generating set <math>a_i, i \in I</math> for a totally ordered indexing set <math>I</math> and relations of the form:
+A '''power-commutator presentation''' of a [[group]] <math>G</math> for a [[prime number]] <math>p</,ath> is a [[presentation]] with generating set <math>a_i, i \in I</math> for a totally ordered indexing set <math>I</math> and relations of the form:
-* ''power relations'': <math>a_i^p</math> is written as a product of powers of <math>a_k, k > i</math>, with the <math>k</math>s in increasing order as we go from left to right in the product.
+* ''power relations'': <math>a_i^p</math> is written as a product of powers of <math>a_k, k > i</math>, with the <math>k</math>s in increasing order as we go from left to right in the product. The exponent on <math>a_k</math> for <math>k > i</math> is denoted <math>\beta(i,k)</math>.
-* ''commutator relations'': The commutator <math>[a_i,a_j]</math> is written as a product of powers of <math>a_k, k > \max \{ i , j \}</math>, with the <math>k</math>s in increasing order as we go from left to right in the product.
+* ''commutator relations'': The commutator <math>[a_i,a_j]</math> is written as a product of powers of <math>a_k, k > \max \{ i , j \}</math>, with the <math>k</math>s in increasing order as we go from left to right in the product. The exponent of <math>a_k</math> for <math>k > \max \{ i , j \}</math> is denoted <math>\beta(i,j,k)</math>.
 For a group of prime power order <math>p^n</math>, a power-commutator presentation is termed ''consistent'' if it uses exactly <math>n</math> generators.