Direct product of cyclic group of prime-square order and cyclic group of prime order
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
|Value of prime number||Corresponding group|
|2||direct product of Z4 and Z2|
|3||direct product of Z9 and Z3|
|5||direct product of Z25 and Z5|
Compare and contrast arithmetic function values with other groups of prime-cube order at Groups of prime-cube order#Arithmetic functions
Arithmetic functions taking values between 0 and 3
Arithmetic functions of a counting nature
Note that since the group is abelian, the number of subgroups equals the number of conjugacy classes of subgroups as well as the number of normal subgroups.
|number of subgroups|
|number of automorphism classes of subgroups||6|
|number of characteristic subgroups||4|
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(p^3,2);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [p^3,2]
or just do:
to have GAP output the group ID, that we can then compare to what we want.