Power-commutator presentation - Revision history

Vipul at 03:48, 1 November 2015

2015-11-01T03:48:41Z

← Older revision		Revision as of 03:48, 1 November 2015
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	==Definition==		==Definition==

	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:		A '''power-commutator presentation''' of a [[group]] <math>G</math> for a [[prime number]] <math>p</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:

	* ''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>.		* ''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>.

Vipul at 03:48, 1 November 2015

2015-11-01T03:48:32Z

← Older revision		Revision as of 03:48, 1 November 2015
Line 1:		Line 1:
	==Definition==		==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.		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.

Vipul at 00:08, 15 December 2011

2011-12-15T00:08:08Z

← Older revision		Revision as of 00:08, 15 December 2011
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	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> 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^~~{r_i}~~</math> ~~where~~ <math>~~r_i~~</math> ~~is a nonnegative integer depending on~~ <math>i</math>.		* ''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.
	* ''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.

Vipul at 03:59, 24 November 2011

2011-11-24T03:59:49Z

← Older revision		Revision as of 03:59, 24 November 2011
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	* ''power relations'': <math>a_i^{r_i}</math> where <math>r_i</math> is a nonnegative integer depending on <math>i</math>.		* ''power relations'': <math>a_i^{r_i}</math> where <math>r_i</math> is a nonnegative integer depending on <math>i</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.

			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.

	==Facts==		==Facts==

	* [[Every group of prime power order has a power-commutator presentation]]		* [[Every group of prime power order has a power-commutator presentation]]

Vipul: Created page with "==Definition== A '''power-commutator presentation''' of a group $G$ is a presentation with generating set $a_i, i \in I$ for a totally ordered inde..."

2011-11-24T03:58:27Z

Created page with "==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 inde..."

New page

==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:

* ''power relations'': <math>a_i^{r_i}</math> where <math>r_i</math> is a nonnegative integer depending on <math>i</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.

==Facts==

* [[Every group of prime power order has a power-commutator presentation]]

Power-commutator presentation - Revision history

Vipul at 03:48, 1 November 2015

Vipul at 03:48, 1 November 2015

Vipul at 00:08, 15 December 2011

Vipul at 03:59, 24 November 2011

Vipul: Created page with "==Definition== A '''power-commutator presentation''' of a group G is a presentation with generating set a_i, i \in I for a totally ordered inde..."

Vipul: Created page with "==Definition== A '''power-commutator presentation''' of a group $G$ is a presentation with generating set $a_i, i \in I$ for a totally ordered inde..."