Group of prime exponent: Difference between revisions

Latest revision as of 01:33, 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
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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 finite 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. Note that for those primes $p$ for which the Burnside problem has an answer of No, it is possible to have an infinite group of exponent $p$ with a finite generating set. However, there will still be many finite groups of interest with exponent $p$ and a finite generating set.

Relation with other properties

Stronger properties

Elementary Abelian group

@@ Line 12: / Line 12: @@
 | 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>
+| 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 finite 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]].
+For related information, see the [[Burnside problem]]. Note that for those primes <math>p</math> for which the Burnside problem has an answer of ''No'', it is possible to have an infinite group of exponent <math>p</math> with a finite generating set. However, there will still be many finite groups of interest with exponent <math>p</math> and a finite generating set.
 ==Relation with other properties==