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# User:Tilman Piesk/Subgroups of powers of Z2

*The sequence was approved: https://oeis.org/A190939**See also: User:Tilman Piesk/Subgroups of powers of Z2/submission form*

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 to , with 0 representing the neutral element.

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

Than every subgroup of Z_{2}^{n} ( )
can be assigned an integer

So for each Z_{2}^{n} there is a finite sequence of these numbers ordered by size.

It is always the beginnig of the finite sequence for Z_{2}^{n+1}.

This leads to the infinite sequence, that starts like this:

- 1, (until here for Z
_{2}^{0}) - 3, (until here for Z
_{2}^{1}) - 5, 9, 15, (until here for Z
_{2}^{2}) - 17, 33, 51, 65, 85, 105, 129, 153, 165, 195, 255, (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, (until here for Z
_{2}^{4}) - 65537, ...

The following file on Wikimedia Commons shows the 67 subgroups of Z_{2}^{4} 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 Z_{2}^{3} 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 Z_{2}^{4} and a Steiner triple system:

File:Jupiter square magic sums vectorial.svg