Arithmetic functions for groups of order 3^n

Distributions for individual arithmetic functions
In the tables here, a row value of $$n$$ means we are looking at the groups of order $$3^n$$. The entry in a cell is the number of isomorphism classes of groups of order $$3^n$$ for which the function takes the value indicated in the column. Note that, for greater visual clarity, all zeros that occur after the last nonzero entry in a row are omitted and the corresponding entry is left blank.

Nilpotency class


Here is the GAP code to generate this information:

We use the function SortArithmeticFunctionSizes, which is not in-built but is easy to code (follow link to get code). We also use the in-built function NilpotencyClassOfGroup. Using these, we get:

gap> SortArithmeticFunctionSizes(3,0,NilpotencyClassOfGroup); [ [ 0, 1 ] ] gap> SortArithmeticFunctionSizes(3,1,NilpotencyClassOfGroup); [ [ 0, 0 ], [ 1, 1 ] ] gap> SortArithmeticFunctionSizes(3,2,NilpotencyClassOfGroup); [ [ 0, 0 ], [ 1, 2 ], [ 2, 0 ] ] gap> SortArithmeticFunctionSizes(3,3,NilpotencyClassOfGroup); [ [ 0, 0 ], [ 1, 3 ], [ 2, 2 ], [ 3, 0 ] ] gap> SortArithmeticFunctionSizes(3,4,NilpotencyClassOfGroup); [ [ 0, 0 ], [ 1, 5 ], [ 2, 6 ], [ 3, 4 ], [ 4, 0 ] ] gap> SortArithmeticFunctionSizes(3,5,NilpotencyClassOfGroup); [ [ 0, 0 ], [ 1, 7 ], [ 2, 28 ], [ 3, 26 ], [ 4, 6 ], [ 5, 0 ] ] gap> SortArithmeticFunctionSizes(3,6,NilpotencyClassOfGroup); [ [ 0, 0 ], [ 1, 11 ], [ 2, 133 ], [ 3, 282 ], [ 4, 71 ], [ 5, 7 ], [ 6, 0 ] ] gap> SortArithmeticFunctionSizes(3,7,NilpotencyClassOfGroup); [ [ 0, 0 ], [ 1, 15 ], [ 2, 1757 ], [ 3, 6050 ], [ 4, 1309 ], [ 5, 173 ], [ 6, 6 ], [ 7, 0 ] ]

Note that for the $$3^7$$ case, the new version of the SmallGroup library (obtain here) is needed, and additional memory may need to be allocated using the command line option `-o' (e.g., gap -o 5G).

Here is the same information, now given in terms of the fraction of groups of a given order that are of a given nilpotency class. For ease of comparison, all fractions are written as decimals, rounded to the fourth decimal place.

Below is information for the probability distribution of nilpotency class using the cohomology tree probability distribution:



Derived length


Here is the GAP code to generate this information:

We use the function SortArithmeticFunctionSizes, which is not in-built but is easy to code (follow link to get code). With this function written, the above data can be generated directly:

gap> SortArithmeticFunctionSizes(3,0,DerivedLength); [ [ 0, 1 ] ] gap> SortArithmeticFunctionSizes(3,1,DerivedLength); [ [ 0, 0 ], [ 1, 1 ] ] gap> SortArithmeticFunctionSizes(3,2,DerivedLength); [ [ 0, 0 ], [ 1, 2 ], [ 2, 0 ] ] gap> SortArithmeticFunctionSizes(3,3,DerivedLength); [ [ 0, 0 ], [ 1, 3 ], [ 2, 2 ], [ 3, 0 ] ] gap> SortArithmeticFunctionSizes(3,4,DerivedLength); [ [ 0, 0 ], [ 1, 5 ], [ 2, 10 ], [ 3, 0 ], [ 4, 0 ] ] gap> SortArithmeticFunctionSizes(3,5,DerivedLength); [ [ 0, 0 ], [ 1, 7 ], [ 2, 60 ], [ 3, 0 ], [ 4, 0 ], [ 5, 0 ] ] gap> SortArithmeticFunctionSizes(3,6,DerivedLength); [ [ 0, 0 ], [ 1, 11 ], [ 2, 493 ], [ 3, 0 ], [ 4, 0 ], [ 5, 0 ], [ 6, 0 ] ] gap> SortArithmeticFunctionSizes(3,7,DerivedLength); [ [ 0, 0 ], [ 1, 15 ], [ 2, 9235 ], [ 3, 60 ], [ 4, 0 ], [ 5, 0 ], [ 6, 0 ], [ 7, 0 ] ]

Note that for the $$3^7$$ case, the new version of the SmallGroup library (obtain here) is needed, and additional memory may need to be allocated using the command line option `-o' (e.g., gap -o 5G).

Here is the same information, now given in terms of the fraction of groups of a given order that are of a given derived length. For ease of comparison, all fractions are written as decimals, rounded to the fourth decimal place.



Frattini length


Here is the GAP code to generate this information:

We use the functions SortArithmeticFunctionSizes and FrattiniLength, which are not in-built but are easy to code (follow link to get code). With this function written, the above data can be generated directly:

gap> SortArithmeticFunctionSizes(3,1,FrattiniLength); [ [ 0, 0 ], [ 1, 1 ] ] gap> SortArithmeticFunctionSizes(3,2,FrattiniLength); [ [ 0, 0 ], [ 1, 1 ], [ 2, 1 ] ] gap> SortArithmeticFunctionSizes(3,3,FrattiniLength); [ [ 0, 0 ], [ 1, 1 ], [ 2, 3 ], [ 3, 1 ] ] gap> SortArithmeticFunctionSizes(3,4,FrattiniLength); [ [ 0, 0 ], [ 1, 1 ], [ 2, 11 ], [ 3, 2 ], [ 4, 1 ] ] gap> SortArithmeticFunctionSizes(3,5,FrattiniLength); [ [ 0, 0 ], [ 1, 1 ], [ 2, 46 ], [ 3, 17 ], [ 4, 2 ], [ 5, 1 ] ] gap> SortArithmeticFunctionSizes(3,6,FrattiniLength); [ [ 0, 0 ], [ 1, 1 ], [ 2, 355 ], [ 3, 133 ], [ 4, 12 ], [ 5, 2 ], [ 6, 1 ] ] gap> SortArithmeticFunctionSizes(3,7,FrattiniLength); [ [ 0, 0 ], [ 1, 1 ], [ 2, 7192 ], [ 3, 2018 ], [ 4, 84 ], [ 5, 12 ], [ 6, 2 ], [ 7, 1 ] ]

Note that for the $$3^7$$ case, the new version of the SmallGroup library (obtain here) is needed, and additional memory may need to be allocated using the command line option `-o' (e.g., gap -o 5G).



Nilpotency class-cum-derived length
Note that in considering the possibilities here, we use the fact that derived length is logarithmically bounded by nilpotency class; explicitly, the derived length is at most $$[\log_2 c] + 1$$ where $$[]$$ is the greatest integer function and $$c$$ is the nilpotency class. On the other hand, derived length gives no upper bound on nilpotency class for derived length at least 2.





Order-cum-power statistics and nilpotency class
The arithmetic function we consider here is the smallest nilpotency class among all groups that are order-cum-power statistics-equvalent to it, i.e., have the same order-cum-power statistics.

Here is the GAP code to generate this information:

The code uses two functions: SortArithmeticFunctionSizes and MinimumNilpotencyClassWithSameOrderCumPowerStatistics, neither of which is in-built (follow links to get code). Using these, we can generate the information as follows:

gap> SortArithmeticFunctionSizes(3,0,MinimumNilpotencyClassWithSameOrderCumPowerStatistics); [ [ 0, 1 ] ] gap> SortArithmeticFunctionSizes(3,1,MinimumNilpotencyClassWithSameOrderCumPowerStatistics); [ [ 0, 0 ], [ 1, 1 ] ] gap> SortArithmeticFunctionSizes(3,2,MinimumNilpotencyClassWithSameOrderCumPowerStatistics); [ [ 0, 0 ], [ 1, 2 ], [ 2, 0 ] ] gap> SortArithmeticFunctionSizes(3,3,MinimumNilpotencyClassWithSameOrderCumPowerStatistics); [ [ 0, 0 ], [ 1, 5 ], [ 2, 0 ], [ 3, 0 ] ] gap> SortArithmeticFunctionSizes(3,4,MinimumNilpotencyClassWithSameOrderCumPowerStatistics); [ [ 0, 0 ], [ 1, 12 ], [ 2, 0 ], [ 3, 3 ], [ 4, 0 ] ] gap> SortArithmeticFunctionSizes(3,5,MinimumNilpotencyClassWithSameOrderCumPowerStatistics); [ [ 0, 0 ], [ 1, 45 ], [ 2, 0 ], [ 3, 16 ], [ 4, 6 ], [ 5, 0 ] ]

In the next table, we give the groups of a given nilpotency class and with the minimum nilpotency class among all groups that are order-cum-power statistics-equivalent to it. Note that because of the Baer correspondence, we know that if the class is $$2$$, then the group is 1-isomorphic to, and hence has the same order-cum-power statistics as, an abelian group. Hence, we remove all the columns with "min-class 2" in them.