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User:Tilman Piesk/Subgroups of powers of Z2

The sequence was approved: https://oeis.org/A190939

Let Z2 be the cyclic group of order 2, and Z2n the n-times product Z2 × ... × Z2. (Like e.g. the Klein four-group Z22.)

The elements of Z2n can be numbered from ${\displaystyle ~0~}$ to ${\displaystyle m=2^{n}-1~}$, with 0 representing the neutral element.

(As all non-neutral elements of Z2n have order 2, this is unambigous: There are no different ways to do so.)

Than every subgroup ${\displaystyle \{0,a,b,c,...\}~}$ of Z2n       ( ${\displaystyle \{0,a,b,c,...\}\subseteq \{0,1,2,...,m\}}$ )       can be assigned an integer ${\displaystyle 1+2^{a}+2^{b}+2^{c}+...~}$

So for each Z2n there is a finite sequence of these numbers ordered by size.
It is always the beginnig of the finite sequence for Z2n+1.
This leads to the infinite sequence, that starts like this:

• 1, (until here for Z20)
• 3, (until here for Z21)
• 5, 9, 15, (until here for Z22)
• 17, 33, 51, 65, 85, 105, 129, 153, 165, 195, 255, (until here for Z23)
• 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, (until here for Z24)
• 65537, ...

The following file on Wikimedia Commons shows the 67 subgroups of Z24 and the corresponding integers:
File:Z2^4; subgroups lexicographical.svg

This sequence has the following subsequences:

According to the number of elements in the subsets, the following subsequences can be defined:

• 4-element subsets:     15, 51, 85, 105, 153, 165, 195, 771, 1285, 1545, 2313, 2565, 3075, 4369, 4641, 5185, 6273, 8481, 8721, 9345, 10305, 12291, 16705, 17025, 17425, 18465, 20485, 24585, 33153, 33345, 33825, 34833, 36873, 40965, 49155, ... (a subsequence of A014312)
• 8-element subsets:     255, 3855, 13107, 15555, 21845, 23205, 26265, 26985, 38505, 39321, 42405, 43605, 50115, 52275, 61455, ... (a subsequence of A023690)
• etc.

Finite Geometry:

There is a bijection between subgroups of Z23 and the Fano plane:
File:Z2^3; Lattice of subgroups Hasse diagram.svg
So probably the whole sequence could also be defined by finite geometry objects, that are generalizations of the Fano plane in higher dimensions.

There is a bijection between the 35 4-element subsets of Z24 and a Steiner triple system:
File:Jupiter square magic sums vectorial.svg