Arithmetic functions for groups of order 2^n
This article gives specific information, namely, arithmetic functions, about a family of groups, namely: groups of order 2^n.
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Contents
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:
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:
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 means we are looking at the groups of order . The entry in a cell is the number of isomorphism classes of groups of order 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.
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 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.
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 |
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).
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
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 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.
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:
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 where is the exponent of the group.
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 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.
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:
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
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 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.
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.
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.
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:
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
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 where is the greatest integer function and is the nilpotency class. On the other hand, derived length gives no upper bound on nilpotency class for derived length at least 2.
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.
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 |
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.
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 |