Group of prime exponent: Difference between revisions

Revision as of 01:32, 5 June 2012

The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter
View other prime-parametrized group properties | View other group properties

Definition

A group of prime exponent is a group whose exponent is a prime number. If $p$ is a prime number, a group of exponent $p$ is a (nontrivial) group in which every element has order $p$ .

Particular cases

Value of prime $p$ ?	What can we say about groups of exponent $p$
2	must be elementary abelian. See exponent two implies abelian
3	must be 2-Engel and class three. Also, if it has a generating set of size $m$ , it must be a quotient of the Burnside group $B (m, 3)$ , which is a finite group of size $3^{m + (\binom{m}{2}) + (\binom{m}{3})}$
5	no bound on nilpotency class. Unknown whether finite generating set forces the group to be finite.

For related information, see the Burnside problem.

Relation with other properties

Stronger properties

Elementary Abelian group

@@ Line 4: / Line 4: @@
 A '''group of prime exponent''' is a group whose [[exponent of a group|exponent]] is a prime number. If <math>p</math> is a [[prime number]], a group of exponent <math>p</math> is a (nontrivial) group in which every element has order <math>p</math>.
+==Particular cases==
+{| class="sortable" border="1"
+! Value of prime <math>p</math>? !! What can we say about groups of exponent <math>p</math>
+|-
+| 2 || must be [[elementary abelian group|elementary abelian]]. See [[exponent two implies abelian]]
+|-
+| 3 || must be [[exponent three implies 2-Engel for groups|2-Engel]] and [[exponent three implies class three|class three]]. Also, if it has a generating set of size <math>m</math>, it must be a quotient of the [[Burnside group]] <math>B(m,3)</math>, which is a finite group of size <math>3^{m + \binom{m}{2} + \binom{m}{3}}</math>
+|-
+| 5 || no bound on nilpotency class. Unknown whether finite generating set forces the group to be finite.
+|}
+For related information, see the [[Burnside problem]].
 ==Relation with other properties==