User:Tilman Piesk/Subgroups of powers of Z2/submission form

NAME

Subgroups of (Z_2)^n interpreted as binary numbers

DATA

1, 3, 5, 9, 15, 17, 33, 51, 65, 85, 105, 129, 153, 165, 195, 255, 257, 513, 771, 1025, 1285, 1545, 2049, 2313, 2565, 3075, 3855, 4097, 4369, 4641, 5185, 6273, 8193, 8481, 8721, 9345, 10305, 12291, 13107, 15555, 16385, 16705, 17025, 17425, 18465, 20485, 21845

OFFSET

1

COMMENTS

Let Z_2 be the cyclic group of order 2, and (Z_2)^n the n-times product Z_2 × ... × Z_2. (Like e.g. the Klein four-group (Z_2)^2.)
The elements of (Z_2)^n can be numbered from 0 to 2^n-1, with 0 representing the neutral element.
(This is unambigous, because all non-neutral elements of the elementary abelian group (Z_2)^n have order 2.)
Than every subgroup {0,a,b,c...} of (Z_2)^n can be assigned an integer 1 + 2^a + 2^b + 2^c + ...
For each (Z_2)^n there is a finite sequence of these numbers ordered by size, and it is the beginning of the finite sequence for (Z_2)^(n+1).
This leads to the infinite sequence:

• 1, (1 until here for (Z_2)^0)
• 3, (2 until here for (Z_2)^1)
• 5, 9, 15, (5 until here for (Z_2)^2)
• 17, 33, 51, 65, 85, 105, 129, 153, 165, 195, 255, (16 until here for (Z_2)^3)
• 257, 513, 771, 1025, 1285, 1545, 2049, 2313, 2565, 3075, 3855, 4097, 4369, 4641, 5185, 6273, 8193, 8481, 8721, 9345, 10305, 12291, 13107, 15555, 16385, 16705, 17025, 17425, 18465, 20485, 21845, 23205, 24585, 26265, 26985, 32769, 33153, 33345, 33825, 34833, 36873, 38505, 39321, 40965, 42405, 43605, 49155, 50115, 52275, 61455, 65535, (67 until here for (Z_2)^4)
• 65537, ...

The number of subgroups of (Z_2)^n is 1, 2, 5, 67, 374, 2825, ... (A006116)

LINKS

The 67 subgroups of (Z_2)^4 and the corresponding integers are shown here:
http://commons.wikimedia.org/wiki/File:Z2%5E4;_subgroups_lexicographical.svg#File

EXAMPLE

The 5 subgroups of the Klein four-group (Z_2)^2 and corresponding integers are:
{0      }     -->     2^0                     =   1
{0,1    }     -->     2^0 + 2^1               =   3
{0,  2  }     -->     2^0       + 2^2         =   5
{0,    3}     -->     2^0             + 2^3   =   9
{0,1,2,3}     -->     2^0 + 2^1 + 2^2 + 2^3   =  15

For the 67 subgroups of (Z_2)^4 and the corresponding integers see section LINKS.

CROSSREFS

Subsequences:

• A051179 2^(2^n) - 1
• A083318 2^n + 1
• A122569 (without the leading 1)

KEYWORD

unkn, tabf

AUTHOR

Tilman Piesk (vimarius(AT)gmail.com)