Derived length distribution of finite p-groups

This article gives both numerical information and links to known facts/conjectures about the distribution of nilpotency class among finite p-groups, i.e., how many p-groups there are of a given derived length.

Related information

• Nilpotency class distribution of finite p-groups
• Nilpotency class-cum-derived length distribution of finite p-groups

Facts known for specific primes

The case $p=2$

$n$	$2^{n}$	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]

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 DerivedLength. Using these functions, the above data can be generated as follows:

gap> SortArithmeticFunctionSizes(2,0,DerivedLength);
[ [ 0, 1 ] ]
gap> SortArithmeticFunctionSizes(2,1,DerivedLength);
[ [ 0, 0 ], [ 1, 1 ] ]
gap> SortArithmeticFunctionSizes(2,2,DerivedLength);
[ [ 0, 0 ], [ 1, 2 ], [ 2, 0 ] ]
gap> SortArithmeticFunctionSizes(2,3,DerivedLength);
[ [ 0, 0 ], [ 1, 3 ], [ 2, 2 ], [ 3, 0 ] ]
gap> SortArithmeticFunctionSizes(2,4,DerivedLength);
[ [ 0, 0 ], [ 1, 5 ], [ 2, 9 ], [ 3, 0 ], [ 4, 0 ] ]
gap> SortArithmeticFunctionSizes(2,5,DerivedLength);
[ [ 0, 0 ], [ 1, 7 ], [ 2, 44 ], [ 3, 0 ], [ 4, 0 ], [ 5, 0 ] ]
gap> SortArithmeticFunctionSizes(2,6,DerivedLength);
[ [ 0, 0 ], [ 1, 11 ], [ 2, 256 ], [ 3, 0 ], [ 4, 0 ], [ 5, 0 ], [ 6, 0 ] ]
gap> SortArithmeticFunctionSizes(2,7,DerivedLength);
[ [ 0, 0 ], [ 1, 15 ], [ 2, 2299 ], [ 3, 14 ], [ 4, 0 ], [ 5, 0 ], [ 6, 0 ], [ 7, 0 ] ]
gap> SortArithmeticFunctionSizes(2,8,DerivedLength);
[ [ 0, 0 ], [ 1, 22 ], [ 2, 55660 ], [ 3, 410 ], [ 4, 0 ], [ 5, 0 ], [ 6, 0 ], [ 7, 0 ], [ 8, 0 ] ]

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

The case $p=5$

$n$	$5^{n}$	total number of groups	length 0	length 1	length 2	length 3
0	1	1	1
1	5	1	0	1
2	25	2	0	2
3	125	5	0	3	2
4	625	15	0	5	10
5	3125	77	0	7	70
6	15625	684	0	11	657	16
7	78125	34297	0	15	33427	855

Here is the GAP code to generate this information: [SHOW MORE]

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 DerivedLength. Using these, we get:

gap> SortArithmeticFunctionSizes(5,0,NilpotencyClassOfGroup);
[ [ 0, 1 ] ]
gap> SortArithmeticFunctionSizes(5,0,DerivedLength);
[ [ 0, 1 ] ]
gap> SortArithmeticFunctionSizes(5,1,DerivedLength);
[ [ 0, 0 ], [ 1, 1 ] ]
gap> SortArithmeticFunctionSizes(5,2,DerivedLength);
[ [ 0, 0 ], [ 1, 2 ], [ 2, 0 ] ]
gap> SortArithmeticFunctionSizes(5,3,DerivedLength);
[ [ 0, 0 ], [ 1, 3 ], [ 2, 2 ], [ 3, 0 ] ]
gap> SortArithmeticFunctionSizes(5,4,DerivedLength);
[ [ 0, 0 ], [ 1, 5 ], [ 2, 10 ], [ 3, 0 ], [ 4, 0 ] ]
gap> SortArithmeticFunctionSizes(5,5,DerivedLength);
[ [ 0, 0 ], [ 1, 7 ], [ 2, 70 ], [ 3, 0 ], [ 4, 0 ], [ 5, 0 ] ]
gap> SortArithmeticFunctionSizes(5,6,DerivedLength);
[ [ 0, 0 ], [ 1, 11 ], [ 2, 657 ], [ 3, 16 ], [ 4, 0 ], [ 5, 0 ], [ 6, 0 ] ]
gap> SortArithmeticFunctionSizes(5,7,DerivedLength);
[ [ 0, 0 ], [ 1, 15 ], [ 2, 33427 ], [ 3, 855 ], [ 4, 0 ], [ 5, 0 ], [ 6, 0 ], [ 7, 0 ] ]

Note that for the $5^{7}$ 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.

$n$	$5^{n}$	total number of groups	length 0	length 1	length 2	length 3
0	1	1	1
1	5	1	0	1
2	25	2	0	1
3	125	5	0	0.6000	0.4000
4	625	15	0	0.3333	0.6667
5	3125	77	0	0.0909	0.9091
6	15625	684	0	0.0161	0.9605	0.0234
7	78125	34297	0	0.0044	0.9746	0.0249

The case $p=7$

$n$	$7^{n}$	total number of groups	length 0	length 1	length 2	length 3
0	1	1	1
1	7	1	0	1
2	49	2	0	2
3	343	5	0	3	2
4	2401	15	0	5	10
5	16807	83	0	7	76
6	117649	860	0	11	829	20

Here is the GAP code to generate this information: [SHOW MORE]

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 DerivedLength. Using these functions, the above data can be generated as follows:

gap> SortArithmeticFunctionSizes(7,1,DerivedLength);
[ [ 0, 0 ], [ 1, 1 ] ]
gap> SortArithmeticFunctionSizes(7,2,DerivedLength);
[ [ 0, 0 ], [ 1, 2 ], [ 2, 0 ] ]
gap> SortArithmeticFunctionSizes(7,3,DerivedLength);
[ [ 0, 0 ], [ 1, 3 ], [ 2, 2 ], [ 3, 0 ] ]
gap> SortArithmeticFunctionSizes(7,4,DerivedLength);
[ [ 0, 0 ], [ 1, 5 ], [ 2, 10 ], [ 3, 0 ], [ 4, 0 ] ]
gap> SortArithmeticFunctionSizes(7,5,DerivedLength);
[ [ 0, 0 ], [ 1, 7 ], [ 2, 76 ], [ 3, 0 ], [ 4, 0 ], [ 5, 0 ] ]
gap> SortArithmeticFunctionSizes(7,6,DerivedLength);
[ [ 0, 0 ], [ 1, 11 ], [ 2, 829 ], [ 3, 20 ], [ 4, 0 ], [ 5, 0 ], [ 6, 0 ] ]

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.

$n$	$7^{n}$	total number of groups	length 0	length 1	length 2	length 3
0	1	1	1
1	7	1	0	1
2	49	2	0	1
3	343	5	0	0.6000	0.4000
4	2401	15	0	0.3333	0.6667
5	16807	83	0	0.0843	0.9157
6	117649	860	0	0.0128	0.9640	0.0233