Burnside group: Difference between revisions
No edit summary |
No edit summary |
||
| Line 9: | Line 9: | ||
The '''Burnside group''' <math>B(n,d)</math> (sometimes called the '''free Burnside group''') is defined as the quotient of the [[free group]] on <math>n</math> generators by the [[normal subgroup]] generated by all <math>d^{th}</math> powers. A Burnside group is a group that occurs as <math>B(n,d)</math> for some choice of <math>d</math> and <math>n</math>. | The '''Burnside group''' <math>B(n,d)</math> (sometimes called the '''free Burnside group''') is defined as the quotient of the [[free group]] on <math>n</math> generators by the [[normal subgroup]] generated by all <math>d^{th}</math> powers. A Burnside group is a group that occurs as <math>B(n,d)</math> for some choice of <math>d</math> and <math>n</math>. | ||
==Particular cases== | |||
{| class="sortable" border="1" | |||
! Value of <math>d</math> !! What can we conclude about <math>B(n,d)</math>? !! Order as a function of <math>n,d</math> !! Nilpotency class in terms of <matH>n,d</math> (assume <math>n > 0</math>) | |||
|- | |||
| 0 || [[finitely generated free group]] on <math>n</matH> generators || infinite || not nilpotent | |||
|- | |||
| 1 || [[trivial group]], regardless of <math>n</math> || 1 || 0 | |||
|- | |||
| 2 || [[elementary abelian group|elementary abelian 2-group]] of rank <math>n</math> and order <math>2^n</math> || <math>2^n</math> || 1 | |||
|- | |||
| 3 || 2-Engel group with <math>n</math> generators, exponent three || <math>3^{n + \binom{n}{2} + \binom{n}{3}}</math> || 1 if <math>n = 1</math><br>2 if <math>n = 2</math><br>3 if <math>n \ge 3</math> | |||
|- | |||
| 4 || {{fillin}} || {{fillin}} || {{fillin}} | |||
|- | |||
| 5 || unknown || unknown, but at least <math>5^6</math> || unknown, but at least 4 if finite | |||
|} | |||
==Relation with other properties== | ==Relation with other properties== | ||
Revision as of 07:28, 20 May 2012
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
This term is related to: combinatorial group theory
View other terms related to combinatorial group theory | View facts related to combinatorial group theory
Definition
Definition with symbols
The Burnside group (sometimes called the free Burnside group) is defined as the quotient of the free group on generators by the normal subgroup generated by all powers. A Burnside group is a group that occurs as for some choice of and .
Particular cases
| Value of | What can we conclude about ? | Order as a function of | Nilpotency class in terms of (assume ) |
|---|---|---|---|
| 0 | finitely generated free group on generators | infinite | not nilpotent |
| 1 | trivial group, regardless of | 1 | 0 |
| 2 | elementary abelian 2-group of rank and order | 1 | |
| 3 | 2-Engel group with generators, exponent three | 1 if 2 if 3 if | |
| 4 | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
| 5 | unknown | unknown, but at least | unknown, but at least 4 if finite |
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Finitely generated free group | Burnside group | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Finitely generated group | ||||
| Reduced free group | |FULL LIST, MORE INFO |