Group of prime power order: Difference between revisions

Latest revision as of 03:35, 17 December 2011

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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This article defines a property that can be evaluated for finite groups (and hence, a particular kind of group property)
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Definition

Symbol-free definition

A group of prime power order is defined in the following equivalent ways:

It is a finite group whose order is a power of a prime.
It is a finite group that is also a p-group for some prime $p$ : the order of every element is a power of that same prime $p$

Equivalence of definitions

For full proof, refer: Equivalence of definitions of group of prime power order

Relation with other properties

Weaker properties

Nilpotent group: For full proof, refer: Prime power order implies nilpotent
Solvable group
Group whose order has at most two prime factors

Specific information

See groups of prime power order for more specific information.

@@ Line 1: / Line 1: @@
+[[importance rank::2| ]]
 {{prime-parametrized group property}}
 {{finite group property}}
 ==Definition==
-A '''group of prime power order''' is defined as a [[finite group]] whose order is a power of a prime.
+===Symbol-free definition===
+A '''group of prime power order''' is defined in the following equivalent ways:
+* It is a [[finite group]] whose order is a power of a prime.
+* It is a [[finite group]] that is also a [[p-group]] for some prime <math>p</math>: the [[order of an element|order]] of every element is a power of that same prime <math>p</math>
+===Equivalence of definitions===
+{{proofat|[[Equivalence of definitions of group of prime power order]]}}
 ==Relation with other properties==
@@ Line 15: / Line 22: @@
 * [[Group whose order has at most two prime factors]]
-==Distribution of orders==
+==Specific information==
-===Groups of prime order===
-For every prime <math>p</math>, there is only one group of order <math>p</math>, viz the cyclic group of <math>p</math> elements.
-===Groups of prime-squared order===
-Any group whose order is the square of a prime must be Abelian. {{proofat|[[Prime squared is Abelianness-forcing]]}}
-Hence there are two possibilities for such a group: the cyclic group of order <math>p^2</math> and the [[elementary Abelian group]] of order <math>p^2</math>.
-===Groups of prime-cubed order===
-For the order a cube of a prime, there are three Abelian possibilities (corresponding to the three possible unordered partitions of 3). There are two non-Abelian possibilities, the [[group of prime-cubed order:U3p|group of unipotent matrices of order 3 over the prime field]] and the semidirect product of the cyclic group of order <math>p^2</math> by a cyclic group of order <math>p</math>.
-===For higher orders===
-{{further|[[enumeration of groups of prime power order]]}}
-The number of groups of order <math>16 = 2^4</math> is 14. For an odd prime <math>p</math>, the number of groups of order <math>p^4</math> is 15.
-For higher powers of the prime, the number of groups of prime power order depends on the prime (in general, the larger the prime, the greater the number of groups).
-Higman and Sims have studied in detail the function:
-<math>f(n,p)</math>
-which measures the number of isomorphism classes of groups of order <math>p^n</math>.
+See [[groups of prime power order]] for more specific information.