Abelian group of prime power order: Difference between revisions

Revision as of 14:59, 18 July 2010

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: abelian group and group of prime power order
View other group property conjunctions OR view all group properties

Definition

An abelian group of prime power order is a group of prime power order that is also an abelian group.

Classification

As a particular case of the structure theorem for finitely generated abelian groups, we can say the following. For a prime $p$ and a nonnegative integer $n$ , the abelian groups of order $p^{n}$ correspond to unordered integer partitions of $n$ . Specifically, a partition $n = k_{1} + k_{2} + \dots + k_{r}$ corresponds to the group:

$\prod_{i = 1}^{r} Z / p_{i}^{k_{i}} Z$

Examples

Partition of $n$	Value of $n$	Case $p = 2$	Case $p = 3$	General case
empty	0	trivial group	trivial group	trivial group
1	1	cyclic group:Z2	cyclic group:Z3	group of prime order
2	2	cyclic group:Z4	cyclic group:Z9	cyclic group of prime-square order
1 + 1	2	Klein four-group	elementary abelian group:E9	elementary abelian group of prime-square order
3	3	cyclic group:Z8	cyclic group:Z27	cyclic group of prime-cube order
2 + 1	3	direct product of Z4 and Z2	direct product of Z9 and Z3	direct product of cyclic group of prime-square order and cyclic group of prime order
1 + 1 + 1	3	elementary abelian group:E8	elementary abelian group:E27	elementary abelian group of prime-cube order
4	4	cyclic group:Z16	cyclic group:Z81	cyclic group of prime-fourth order
3 + 1	4	direct product of Z8 and Z2	direct product of Z27 and Z3	direct product of cyclic group of prime-cube order and cyclic group of prime order
2 + 2	4	direct product of Z4 and Z4	direct product of Z9 and Z9	direct product of cyclic group of prime-square order and cyclic group of prime-square order
2 + 1 + 1	4	direct product of Z4 and V4	direct product Z9 and E9	direct product of cyclic group of prime-square order and elementary abelian group of prime-square order
1 + 1 + 1 + 1	4	elementary abelian group:E16	elementary abelian group:E81	elementary abelian group of prime-fourth order

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
cyclic group of prime power order	the corresponding partition has just one piece
homocyclic group of prime power order	the corresponding partition has all pieces of equal size
Finite elementary abelian group	the corresponding partition has all pieces of size 1

@@ Line 5: / Line 5: @@
 An '''abelian group of prime power order''' is a [[group of prime power order]] that is also an [[abelian group]].
+==Classification==
+As a particular case of the [[structure theorem for finitely generated abelian groups]], we can say the following. For a prime <math>p</math> and a nonnegative integer <math>n</math>, the abelian groups of order <math>p^n</math> correspond to [[set of unordered integer partitions|unordered integer partitions]] of <math>n</math>. Specifically, a partition <math>n = k_1 + k_2 + \dots + k_r</math> corresponds to the group:
+<math>\! \prod_{i=1}^r \mathbb{Z}/p_i^{k_i}\mathbb{Z}</math>
+==Examples==
+{| class="sortable" border="1"
+! Partition of <math>n</math> !! Value of <math>n</math> !! Case <math>p = 2</math> !! Case <math>p = 3</math> !! General case
+|-
+| empty || 0 || [[trivial group]] || [[trivial group]] || [[trivial group]]
+|-
+| 1 || 1 || [[cyclic group:Z2]] || [[cyclic group:Z3]] || [[group of prime order]]
+|-
+| 2 || 2 || [[cyclic group:Z4]] || [[cyclic group:Z9]] || [[cyclic group of prime-square order]]
+|-
+| 1 + 1 || 2 || [[Klein four-group]] || [[elementary abelian group:E9]] || [[elementary abelian group of prime-square order]]
+|-
+| 3 || 3 || [[cyclic group:Z8]] || [[cyclic group:Z27]] || [[cyclic group of prime-cube order]]
+|-
+| 2 + 1 || 3 || [[direct product of Z4 and Z2]] || [[direct product of Z9 and Z3]] || [[direct product of cyclic group of prime-square order and cyclic group of prime order]]
+|-
+| 1 + 1 + 1 || 3 || [[elementary abelian group:E8]] || [[elementary abelian group:E27]] || [[elementary abelian group of prime-cube order]]
+|-
+| 4 || 4 || [[cyclic group:Z16]] || [[cyclic group:Z81]] || [[cyclic group of prime-fourth order]]
+|-
+| 3 + 1 || 4 || [[direct product of Z8 and Z2]] || [[direct product of Z27 and Z3]] || [[direct product of cyclic group of prime-cube order and cyclic group of prime order]]
+|-
+| 2 + 2 || 4 || [[direct product of Z4 and Z4]] || [[direct product of Z9 and Z9]] || [[direct product of cyclic group of prime-square order and cyclic group of prime-square order]]
+|-
+| 2 + 1 + 1 || 4 || [[direct product of Z4 and V4]] || [[direct product Z9 and E9]] || [[direct product of cyclic group of prime-square order and elementary abelian group of prime-square order]]
+|-
+| 1 + 1 + 1 + 1 || 4 || [[elementary abelian group:E16]] || [[elementary abelian group:E81]] || [[elementary abelian group of prime-fourth order]]
+|}
 ==Relation with other properties==
 ===Stronger properties===
-* [[Weaker than::Cyclic group of prime power order]]
+{| class="sortable" border="1"
-* [[Weaker than::Homocyclic group of prime power order]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Weaker than::Finite elementary abelian group]]
+|-
+| [[Weaker than::cyclic group of prime power order]] || the corresponding partition has just one piece || || ||
+|-
+| [[Weaker than::homocyclic group of prime power order]] || the corresponding partition has all pieces of equal size || || ||
+|-
+| [[Weaker than::Finite elementary abelian group]] || the corresponding partition has all pieces of size 1|| || ||
+|}