Arithmetic functions for groups of order 2^n

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This article gives specific information, namely, arithmetic functions, about a family of groups, namely: groups of order 2^n.
View arithmetic functions for group families | View other specific information about groups of order 2^n

Summary


Below is a summary of the behavior of the average values for important arithmetic functions, where the average is computed by equally weighting all isomorphism classes of groups of that order:

n 2^n number of groups nilpotency class derived length prime-base logarithm of exponent Frattini length minimum size of generating set
0 1 1 0 0 0 0 0
1 2 1 1 1 1 1 1
2 4 2 1 1 1.5 1.5 1.5
3 8 5 1.4 1.4 2 2 2
4 16 14 1.8571 1.6429 2.4286 2.4286 2.3571
5 32 51 2.2745 1.8627 2.6471 2.6471 2.7451
6 64 267 2.5843 1.9588 2.7828 2.7903 3.1124
7 128 2328 2.6937 1.9996 2.7637 2.7668 3.5833
8 256 56092 2.4941 2.0069  ?  ?  ?

Below is a summary of the behavior of the average values where the groups are weighted by the cohomology tree probability distribution:

n 2^n number of groups nilpotency class derived length prime-base logarithm of exponent Frattini length minimum size of generating set
0 1 1 0 0 0 0 0
1 2 1 1 1 1 1 1
2 4 2 1 1 1.5 1.5 1.5
3 8 5 1.25 1.25 2.1875  ? 1.8125
4 16 14 1.6172 1.5234 2.6865  ? 2.0322
5 32 51 1.9889 1.7728 3.1426 ? 2.2039
6 64 267 2.3329 1.8467 3.5342  ? 2.3288


Nilpotency class

Up to isomorphism

In the table here, a row value of n means we are looking at the groups of order 2^n. The entry in a cell is the number of isomorphism classes of groups of order 2^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.


n 2^n total number of groups class 0 class 1 class 2 class 3 class 4 class 5 class 6 class 7
0 1 1 1
1 2 1 0 1
2 4 2 0 2
3 8 5 0 3 2
4 16 14 0 5 6 3
5 32 51 0 7 26 15 3
6 64 267 0 11 117 114 22 3
7 128 2328 0 15 947 1137 197 29 3
8 256 56092 0 22 31742 21325 2642 320 38 3
Here is the GAP code to generate this information: [SHOW MORE]

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.

n 2^n total number of groups average of values (equal weighting on all groups) class 0 class 1 class 2 class 3 class 4 class 5 class 6 class 7
0 1 1 0 1
1 2 1 1 0 1
2 4 2 1 0 1
3 8 5 1.4 0 0.6000 0.4000
4 16 14 1.8571 0 0.3571 0.4286 0.2143
5 32 51 2.2745 0 0.1373 0.5098 0.2941 0.0588
6 64 267 2.5843 0 0.0412 0.4382 0.4270 0.0824 0.0112
7 128 2328 2.6937 0 0.0064 0.4068 0.4884 0.0846 0.01245 0.0013
8 256 56092 2.4941 0 0.0004 0.5659 0.3802 0.0471 0.0057 0.0007 0.0001
Below is the information for the probability distribution by nilpotency class using the cohomology tree probability distribution: [SHOW MORE]


Up to isoclinism

We give below information on the number of equivalence classes under the equivalence relation of being isoclinic groups, for each nilpotency class. The equivalence classes under isoclinism are also called Hall-Senior families. For visual clarity, the cells with zero entries are omitted. We know that isoclinic groups have same nilpotency class, with the exception of the trivial group (which has class zero) being isoclinic to nontrivial abelian groups (which have class 1).

n 2^n total number of equivalence classes under isoclinism class 0 class 1 class 2 class 3 class 4 class 5 class 6 class 7
0 1 1 1
1 2 1 0 1
2 4 1 0 1
3 8 2 0 1 1
4 16 3 0 1 1 1
5 32 8 0 1 3 3 1
6 64 27 0 1  ?  ?  ? 1
7 128 115 0 1  ?  ?  ?  ? 1

Derived length


n 2^n total number of groups length 0 length 1 length 2 length 3
0 1 1 1
1 2 1 0 1
2 4 2 0 2
3 8 5 0 3 2
4 16 14 0 5 9
5 32 51 0 7 44
6 64 267 0 11 256
7 128 2328 0 15 2299 14
8 256 56092 0 22 55660 410
Here is the GAP code to generate this information: [SHOW MORE]

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.

n 2^n total number of groups average of values (equal weighting on all groups) length 0 length 1 length 2 length 3
0 1 1 0 1
1 2 1 1 0 1
2 4 2 1 0 1
3 8 5 1.4 0 0.6000 0.4000
4 16 14 1.6429 0 0.3571 0.6429
5 32 51 1.8627 0 0.1373 0.8627
6 64 267 1.9588 0 0.0412 0.9588
7 128 2328 1.9996 0 0.0064 0.9875 0.0060
8 256 56092 2.0069 0 0.0004 0.9923 0.0073

Below is information for the probability distribution of derived length under the cohomology tree probability distribution:

n 2^n total number of groups average of values (cohomology tree probability distribution) length 0 length 1 length 2 length 3
0 1 1 0 1
1 2 1 1 0 1
2 4 2 1 0 1
3 8 5 1.25 0 0.7500 0.2500
4 16 14 1.5234 0 0.4766 0.5234
5 32 51 1.7228 0 0.2772 0.7228
6 64 267 1.8467 0 0.1533 0.8467


Prime-base logarithm of exponent


The prime-base logarithm of exponent, in this case, is \log_2E where E is the exponent of the group.

n 2^n number of groups value 0 (exponent 1) value 1 (exponent 2) value 2 (exponent 4) value 3 (exponent 8) value 4 (exponent 16) value 5 (exponent 32) value 6 (exponent 64) value 7 (exponent 128)
0 1 1 1
1 2 1 0 1
2 4 2 0 1 1
3 8 5 0 1 3 1
4 16 14 0 1 7 5 1
5 32 51 0 1 23 21 5 1
6 64 267 0 1 96 137 27 5 1
7 128 2328 0 1 823 1269 202 27 5 1
Here is the GAP code to generate this information: [SHOW MORE]

Here is the same information, now given in terms of the fraction of groups of a given order that have a given prime-base logarithm of exponent. For ease of comparison, all fractions are written as decimals, rounded to the fourth decimal place.

n 2^n number of groups average of values (equal weighting on all groups) value 0 value 1 value 2 value 3 value 4 value 5 value 6 value 7
0 1 1 0 1
1 2 1 1 0 1
2 4 2 1.5 0 0.5000 0.5000
3 8 5 2 0 0.2000 0.6000 0.2000
4 16 14 2.4286 0 0.0714 0.5000 0.3571 0.0714
5 32 51 2.6471 0 0.0196 0.4510 0.4118 0.0980 0.0196
6 64 267 2.7828 0 0.0037 0.3596 0.5131 0.1011 0.0187 0.0037
7 128 2328 2.7637 0 0.0004 0.3535 0.5451 0.0868 0.0116 0.0021 0.0004

Below is information on the probability distribution of prime-base logarithm of exponent under the cohomology tree probability distribution:

n 2^n number of groups average of values (cohomology tree probability distribution) value 0 value 1 value 2 value 3 value 4 value 5 value 6 value 7
0 1 1 0 1
1 2 1 1 0 1
2 4 2 1.5 0 0.5000 0.5000
3 8 5 2.1875 0 0.0625 0.6875 0.0250
4 16 14 2.6865 0 0.0010 0.4365 0.4375 0.1250
5 32 51 3.1426 0 0.0000 0.1738 0.5723 0.1914 0.0625
6 64 267 3.5342 0 0.0000 0.0626 0.4963 0.3166 0.0933 0.0313


Frattini length


n 2^n total number of groups length 0 length 1 length 2 length 3 length 4 length 5 length 6 length 7
0 1 1 1
1 2 1 0 1
2 4 2 0 1 1
3 8 5 0 1 3 1
4 16 14 0 1 7 5 1
5 32 51 0 1 23 21 5 1
6 64 267 0 1 94 139 27 5 1
7 128 2328 0 1 816 1276 202 27 5 1
Here is the GAP code to generate this information: [SHOW MORE]

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

n 2^n total number of groups average of values (equal weighting on all groups) length 0 length 1 length 2 length 3 length 4 length 5 length 6 length 7
0 1 1 0 1
1 2 1 1 0 1
2 4 2 1.5 0 0.5000 0.5000
3 8 5 2 0 0.2000 0.6000 0.2000
4 16 14 2.4286 0 0.0714 0.5000 0.3571 0.0714
5 32 51 2.6471 0 0.0196 0.4510 0.4118 0.0980 0.0196
6 64 267 2.7903 0 0.0037 0.3521 0.5206 0.1011 0.0187 0.0037
7 128 2328 2.7668 0 0.0004 0.3505 0.5481 0.0868 0.0116 0.0021 0.0004


Minimum size of generating set


This is also equal to the prime-base logarithm of the order of the Frattini quotient.

n 2^n total number of groups size 0 size 1 size 2 size 3 size 4 size 5 size 6 size 7
0 1 1 1
1 2 1 0 1
2 4 2 0 1 1
3 8 5 0 1 3 1
4 16 14 0 1 8 4 1
5 32 51 0 1 19 24 6 1
6 64 267 0 1 53 137 68 7 1
7 128 2328 0 1 162 833 1153 169 9 1

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

n 2^n total number of groups average value of minimum size of generating set size 0 size 1 size 2 size 3 size 4 size 5 size 6 size 7
0 1 1 0 1
1 2 1 1 0 1
2 4 2 1.5 0 0.5000 0.5000
3 8 5 2 0 0.2000 0.6000 0.2000
4 16 14 2.3571 0 0.0714 0.5714 0.2857 0.0714
5 32 51 2.7451 0 1 19 24 6 1
6 64 267 3.1124 0 1 53 137 68 7 1
7 128 2328 3.5833 0 1 162 833 1153 169 9 1


Below is information on the probability distribution of minimum size of generating set under the cohomology tree probability distribution:

n 2^n total number of groups average of values of minimum size of generating set (by cohomology tree probability distribution) size 0 size 1 size 2 size 3 size 4 size 5 size 6 size 7
0 1 1 0 1
1 2 2 1 0 1
2 4 2 1.5 0 0.5000 0.5000
3 8 5 1.8125 0 0.2500 0.6875 0.0625
4 16 14 2.0322 0 0.1250 0.7188 0.1553 0.0010
5 32 51 2.2039 0 0.0625 0.6753 0.2580 0.0042 0.0000
6 64 267 2.3288 0 0.0313 0.6195 0.3384 0.0011 0.0000 0.0000


Rank of a p-group


n 2^n total number of groups rank 0 rank 1 rank 2 rank 3 rank 4 rank 5 rank 6
0 1 1 1
1 2 1 0 1
2 4 2 0 1 1
3 8 5 0 2 2 1
4 16 14 0 2 8 3 1
5 32 51 0 2 21 23 4 1
6 64 267 0 2 54 150 55 5 1


Interaction of multiple arithmetic functions

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.


n 2^n total number of groups class and length 0 class and length 1 class 2, length 2 class 3, length 2 class 4, length 2 class 4, length 3 class 5, length 2 class 5, length 3 class 6, length 2 class 6, length 3
0 1 1 1
1 2 1 0 1
2 4 2 0 2
3 8 5 0 3 2
4 16 14 0 5 6 3
5 32 51 0 7 26 15 3
6 64 267 0 11 117 114 22 0 3
7 128 2328 0 15 947 1137 187 10 25 4 3


Element structure 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.

n 2^n total number of groups min-class 0 min-class 1 min-class 2 min-class 3 min-class 4 min-class 5 min-class 6
0 1 1 1
1 2 1 0 1
2 4 2 0 2
3 8 5 0 3 2
4 16 14 0 7 4 3
5 32 51 0 15 18 15 3
6 64 267 0 44 95 103 22 3
7 128 2328 0  ?  ?  ?  ?  ? 3
Here is the GAP code to generate this information: [SHOW MORE]

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.

n 2^n total number of groups class and min-class 0 class and min-class 1 class 2, min-class 2 class 2, min-class 1 class 3, min-class 3 class 3, min-class 2 class 3, min-class 1 class 4, min-class 4 class 4, min-class 3 class 4, min-class 2 class 4, min-class 1 class 5, min-class 5
0 1 1 1
1 2 1 0 1
2 4 2 0 2
3 8 5 0 3 2
4 16 14 0 5 4 2 3
5 32 51 0 7 18 8 15 0 0 3
6 64 267 0 11 85 32 103 10 1 22 0 0 0 3