Direct product of cyclic group of prime-square order and cyclic group of prime-square order

From Groupprops
Revision as of 16:20, 18 July 2010 by Vipul (talk | contribs) (Created page with '{{prime-parametrized particular group}} ==Definition== Let <math>p</math> be a prime number. This group is defined in the following equivalent ways: # It is the [[defi...')
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
This article is about a family of groups with a parameter that is prime. For any fixed value of the prime, we get a particular group.
View other such prime-parametrized groups

Definition

Let p be a prime number. This group is defined in the following equivalent ways:

  1. It is the external direct product of two copies of the cyclic group of prime-square order, i.e., it is the group \mathbb{Z}_{p^2} \times \mathbb{Z}_{p^2}.
  2. It is the homocyclic group of order p^4 and exponent p^2.

Particular cases

Value of prime number p Corresponding group
2 direct product of Z4 and Z4
3 direct product of Z9 and Z9
5 direct product of Z25 and Z25

Arithmetic functions

Function Value Similar groups Explanation for function value
prime-base logarithm of order 4 groups with same prime-base logarithm of order
max-length of a group 4 max-length of a group equals prime-base logarithm of order for group of prime power order
chief length 4 chief length equals prime-base logarithm of order for group of prime power order
composition length 4 composition length equals prime-base logarithm of order for group of prime power order
prime-base logarithm of exponent 2 groups with same prime-base logarithm of order and prime-base logarithm of exponent | groups with same prime-base logarithm of exponent
Frattini length 2 groups with same prime-base logarithm of order and Frattini length | groups with same Frattini length Frattini length equals prime-base logarithm of exponent for abelian group of prime power order
minimum size of generating set 2 groups with same prime-base logarithm of order and minimum size of generating set | groups with same minimum size of generating set
subgroup rank of a group 2 groups with same prime-base logarithm of order and subgroup rank of a group | groups with same subgroup rank of a group same as minimum size of generating set since it is an abelian group of prime power order
rank of a p-group 2 groups with same prime-base logarithm of order and rank of a p-group | groups with same rank of a p-group same as minimum size of generating set since it is an abelian group of prime power order
normal rank of a p-group 2 groups with same prime-base logarithm of order and normal rank of a p-group | groups with same normal rank of a p-group same as minimum size of generating set since it is an abelian group of prime power order
characteristic rank of a p-group 2 groups with same prime-base logarithm of order and characteristic rank of a p-group | groups with same characteristic rank of a p-group same as minimum size of generating set since it is an abelian group of prime power order
nilpotency class 1 The group is a nontrivial abelian group
derived length 1 The group is a nontrivial abelian group
Fitting length 1 The group is a nontrivial abelian group

GAP implementation

Group ID

This finite group has order p^4
"p^" can not be assigned to a declared number type with value 4.
and has ID 2 among the groups of order p^4 in GAP's SmallGroup library. For context, there are groups of order p^4. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(p^4,2)

For instance, we can use the following assignment in GAP to create the group and name it G:

gap> G := SmallGroup(p^4,2);

Conversely, to check whether a given group G is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [p^4,2]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

"p^" can not be assigned to a declared number type with value 4.

Short descriptions

Description Functions used Mathematical comments
DirectProduct(CyclicGroup(p^2),CyclicGroup(p^2)) DirectProduct and CyclicGroup