The Magic Encyclopedia ™

Group Theory
(by Aale de Winkel)

<Francis Gaspalou> discusses the grouptheoretical aspects of the magic squares in magic-squares
This file list some results based on my own investigation based on <Gil Lamb>'s Generating Squares (see elsewhere in this Encyclopedia)
Dynamic numbering forms a groupoperator with the generating squares as groupgenerators. The order of a group is the amount of elements within the group, this becomes a bit confusing with the regular meaning of the word 'order', so I'll use 'grouporder' where the order of the group is meant.
Most elements of the GS-groups here when dynamically numbered with repeatedly itself will map onto the identity N_m (as I currently understand nomenclature) this forms an (of course) abelean (single generator) DN-subgroup of grouporder p where: GS^p = N_m (below mentioned as: power p)
(note: I said 'most' since I cut off my routine at p = 1000, and order 12 elements I found which do not seem to display the feature or is well beyond this cutoff element like: ₃N₁*₂N₆*N₂)

Order 4 {2-pan} squares
Aside from the 48 {pan}magic squares there are 384 (8*48) {2-pan}magic squares
The 3 order 4 Generating Squares {N₄,₂N₁*₂N₄,₂N₂*₂N₂}
forms a dynamic numbering group of 'group-order 6' (ie 6 elements 'GS')
which combines any of the {2-pan} squares S with 5 others {GS}S
elements {GS}({GS}S) are elements that are already known
Further the 'panrelocation' @[2,1] combined with the "row-panflip' _2[0,3,2,1]
(ie @[2,1]_2[0,3,2,1]) traerses the set of squares in four steps, and can be
combined with the 'diagonal-panflip' _3[0,3,2,1] into a group of 'grouporder 8'
Combining the GS-group with the 'panflip-group' combines into a "grouporder 48'
group {GS} . {@[2,1]_2[0,3,2,1]} . {_3[0,3,2,1]}

N₄
(power: 1)

00 01 02 03
04 05 06 07
08 09 10 11
12 13 14 15 ₂N₁*₂N₄
(power: 3)

00 01 08 09
02 03 10 11
04 05 12 13
06 07 14 15 N₂*N₂
(power: 2)

00 01 04 05
02 03 06 07
08 09 12 13
10 11 14 15

{₂N₁*₂N₄}(₂N₁*₂N₄)
(power: 3)

00 01 04 05
08 09 12 13
02 03 06 07
10 11 14 15 {N₂*N₂}(₂N₁*₂N₄)
(power: 2)

00 01 08 09
04 05 12 13
02 03 10 11
06 07 14 15 {₂N₁*₂N₄}(N₂*N₂)
(power: 2)

00 01 02 03
08 09 10 11
04 05 06 07
12 13 14 15

Further of course
(@[2,1]_2[0,3,2,1)⁴ = @[0,0]_2[0,1,2,3] and
(_3[0,3,2,1])² = _3[0,1,2,3]

The forementioned groupgenerators form a seperation of the 384 {2-pan} squares in
8 seperate groups the frirst elements are squares 16, 18, 21, 27, 30, 32, 34 and 73
as said also the {pan}squares are interconnected by this same group with square 102
as the first element in the database uploaded ordered listing, though being a group
any square within the group can be used and the square "Pan" forms the easiest
constructable group element of the {pan}magic squares by the "Pan-transform".

Order 8 Dynamic numbering group
The 'grouporder' of the group with 10 generators is yet undertermined.

N₈ (power: 1)
₂N₁*₄N₈ (power: 4)
₄N₁*₂N₈ (power: 5)
N₂*N₄ (power: 3)
₄N₂*₂N₄ (power: 2)
₂N₁*N₂*₂N₄ (power: 6)
₂N₄*₄N₂ (power: 2)
N₄*N₂ (power: 3)
₂N₁*₂N₄*N₂ (power: 5)
N₂*N₂*N₂ (power: 4)

The 100 elements {GS}GS has 21 known elements since
{N₈}GS = {GS}N₈ = GS and

{N₂*N₄}(₄N₂*₂N₄) = N₄*N₂
{N₂*N₄}(₂N₁*₂N₄*N₂) = ₄N₁*₂N₈
{₄N₂*₂N₄}(₄N₂*₂N₄*N₂) = N₈
{₂N₄*₄N₂}(₄N₂*₂N₄) = N₂*N₂*N₂
{₂N₄*₄N₂}(₂N₁*N₂*₂N₄) = ₂N₁*₂N₄*N₂
{₂N₄*₄N₂}(₂N₄*₄N₂) = N₈
{₂N₄*₄N₂}(₂N₁*₂N₁*₂N₄*N₂) = ₂N₁*N₂*₂N₄
{₂N₄*₄N₂}(N₂*N₂*N₂) = ₄N₂*₂N₄
{N₄*N₂}(₄N₂*₂N₄) = N₂*N₄
{N₄*N₂}(₂N₁*N₂*₂N₄) = ₂N₁*₄N₈
{N₂*N₂*N₂}(₄N₂*₂N₄) = ₂N₄*₄N₂

the rest are all new elements of the group ("power" ranges from 1 till 7)

Application on the square "Pan" of elements of the above mentioned order 8 DN-group
reveiled only {pan}magic squares, a strong indication that this order 8 group is
invariant for the {panmagic} qualification, it is NOT a {bimagic} invariant.
Only a few elements dynamically numbered the tested bimagic square into a bimagic square
since this did not continue with reapplication of the same element this doen't count

Order 9 Dynamic numbering group
<Gil Lamb> discovered the dynamically numbering the order 9 Collison bimagic square he obtained
yet another bimagic square, this seems to continue for all order 9 bimagic squares.
and not only the {3-pan bimagic 3-panmagic} squares
the Collison square is: (square #184)_{_3[1,6,3,0,4,8,5,2,7]}
LP({{1,1,0,2},{2,1,1,0}},{{0,1,1,2}{2,0,1,1}}_{_[7,2,3,5,6,1,0,4,8]}^t) _{_3[1,6,3,0,4,8,5,2,7]}

The 'grouporder' of this group is the same as for order 4 thus 6 elements:

N₉
₃N₁*₃N₉
N₃*N₃
{₃N₁*₃N₉}(₃N₁*₃N₉)
{₃N₁*₃N₉}(N₃*N₃)
{N₃*N₃}(₃N₁*₃N₉)

Order 12 Dynamic numbering group
The vast order 12 DN-group with 41 generators shows elements that are not {panmagic} invariants
most elements seem to be invariants, a subgroup might be definable that is {panmagic} invariant

Currently I have no idea why the order 9 DN-group works for the bimagic squares, also the thus found {3-pan bimagic 3-panmagic} squares seems not to be obtainable by the tri-digital equations with which I obtained the database listed squares. As above metioned only a few elements worked on the order 8 {4-pan bimagic panmagic} square I tested. The groups seem to be a {panmgic} invariant, though the order 12 evidence indicates tat one might need to exclude some elements as {{₂N₁*₆N₁₂}(N₃*N₄)} Pan is not a panmagic square among others

from Small_Latin_squares_and_quasigroups I learned that there are but 4 reduced latin squares

Order 4 group theory
RLS_A

abcd 0123
badc 1032
cdab 2301
dcba 3210 RLS_B

abcd 0123
badc 1032
cdba 2310
dcab 3201 RLS_C

abcd 0123
bcda 1230
cdab 2301
dabc 3012 RLS_D

abcd 0123
bdac 1302
cadb 2031
dcba 3210
which expands to all 576 possibilities by performing 6 row permutations of type _2[0,perm(1,2,3)]
and the 24 column permutations _1[perm(0,1,2,3)], further associating a,b,c,d with 0,1,2,3 gives
the normally used form of latin squares on this site also shown above
examination learns only RLS_A used in order 4 {panmagic) squares
RLS_A_{_2[0,2,1,3]}

0123
2301
1032
3210 RLS_A_{_2[0,1,3,2]=[0,3,1,2]}

0312
3021
2130
1203 RLS_A_{_2[0,3,1,2]}

0123
3210
1032
2301 RLS_A_{_2[0,2,3,1]=[0,3,1,2]}

0312
1203
2130
3021

Note that the latter 2 are related to the first 2 by the vertical "panflip" _2[0,3,2,1]
while of course the first 2 are related by the combined _2[0,2,3,1]=[0,3,1,2] (!?)
also see that RLS_A_{_2[0,3,1,2]} = RLS_A_{_2[0,2,3,1]=[0,3,1,2]}^t making the
first a bit more natural then the latter since 4 * RLS_A_{_2[0,3,1,2]} + RLS_A_{_2[0,3,1,2]}^t is
in it's normalized position immediately.
As this LS has no equal elements symmetric to the maindiagonal
it is orthogonal to it's transposed and thus all numbers are present in this square
(the square thus obtained is #104 in my uploaded list which is but the vertical panflip
of the first panmagic square (ie #104 = #102_{_2[0,3,2,1]}) which serves as entrypoint for
the dynamic numbering-panflip group of grouporder 48 mentioned in the previous section)

Note: the table above is a novel approach, it will be augmented as time permits

Order 4 {2-pan} squares
Aside from the 48 {pan}magic squares there are 384 (848) {2-pan}magic squares The 3 order 4 Generating Squares {N₄,₂N₁₂N₄,₂N₂*₂N₂} forms a dynamic numbering group of 'group-order 6' (ie 6 elements 'GS') which combines any of the {2-pan} squares S with 5 others {GS}S elements {GS}({GS}S) are elements that are already known Further the 'panrelocation' @[2,1] combined with the "row-panflip' _2[0,3,2,1] (ie @[2,1]_2[0,3,2,1]) traerses the set of squares in four steps, and can be combined with the 'diagonal-panflip' _3[0,3,2,1] into a group of 'grouporder 8' Combining the GS-group with the 'panflip-group' combines into a "grouporder 48' group {GS} . {@[2,1]_2[0,3,2,1]} . {_3[0,3,2,1]}
N₄ (power: 1) 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15	₂N₁*₂N₄ (power: 3) 00 01 08 09 02 03 10 11 04 05 12 13 06 07 14 15	N₂*N₂ (power: 2) 00 01 04 05 02 03 06 07 08 09 12 13 10 11 14 15
{₂N₁₂N₄}(₂N₁₂N₄) (power: 3) 00 01 04 05 08 09 12 13 02 03 06 07 10 11 14 15	{N₂N₂}(₂N₁₂N₄) (power: 2) 00 01 08 09 04 05 12 13 02 03 10 11 06 07 14 15	{₂N₁₂N₄}(N₂N₂) (power: 2) 00 01 02 03 08 09 10 11 04 05 06 07 12 13 14 15
Further of course (@[2,1]_2[0,3,2,1)⁴ = @[0,0]_2[0,1,2,3] and (_3[0,3,2,1])² = _3[0,1,2,3] The forementioned groupgenerators form a seperation of the 384 {2-pan} squares in 8 seperate groups the frirst elements are squares 16, 18, 21, 27, 30, 32, 34 and 73 as said also the {pan}squares are interconnected by this same group with square 102 as the first element in the database uploaded ordered listing, though being a group any square within the group can be used and the square "Pan" forms the easiest constructable group element of the {pan}magic squares by the "Pan-transform".
Order 8 Dynamic numbering group
The 'grouporder' of the group with 10 generators is yet undertermined. N₈ (power: 1) ₂N₁₄N₈ (power: 4) ₄N₁₂N₈ (power: 5) N₂N₄ (power: 3) ₄N₂₂N₄ (power: 2) ₂N₁N₂₂N₄ (power: 6) ₂N₄₄N₂ (power: 2) N₄N₂ (power: 3) ₂N₁₂N₄N₂ (power: 5) N₂N₂N₂ (power: 4)
The 100 elements {GS}GS has 21 known elements since {N₈}GS = {GS}N₈ = GS and {N₂N₄}(₄N₂₂N₄) = N₄N₂ {N₂N₄}(₂N₁₂N₄N₂) = ₄N₁₂N₈ {₄N₂₂N₄}(₄N₂₂N₄N₂) = N₈ {₂N₄₄N₂}(₄N₂₂N₄) = N₂N₂N₂ {₂N₄₄N₂}(₂N₁N₂₂N₄) = ₂N₁₂N₄N₂ {₂N₄₄N₂}(₂N₄₄N₂) = N₈ {₂N₄₄N₂}(₂N₁₂N₁₂N₄N₂) = ₂N₁N₂₂N₄ {₂N₄₄N₂}(N₂N₂N₂) = ₄N₂₂N₄ {N₄N₂}(₄N₂₂N₄) = N₂N₄ {N₄N₂}(₂N₁N₂₂N₄) = ₂N₁₄N₈ {N₂N₂N₂}(₄N₂₂N₄) = ₂N₄₄N₂ the rest are all new elements of the group ("power" ranges from 1 till 7)
Application on the square "Pan" of elements of the above mentioned order 8 DN-group reveiled only {pan}magic squares, a strong indication that this order 8 group is invariant for the {panmagic} qualification, it is NOT a {bimagic} invariant. Only a few elements dynamically numbered the tested bimagic square into a bimagic square since this did not continue with reapplication of the same element this doen't count
Order 9 Dynamic numbering group
<Gil Lamb> discovered the dynamically numbering the order 9 Collison bimagic square he obtained yet another bimagic square, this seems to continue for all order 9 bimagic squares. and not only the {3-pan bimagic 3-panmagic} squares the Collison square is: (square #184)_{_3[1,6,3,0,4,8,5,2,7]} LP({{1,1,0,2},{2,1,1,0}},{{0,1,1,2}{2,0,1,1}}_{_[7,2,3,5,6,1,0,4,8]}^t) _{_3[1,6,3,0,4,8,5,2,7]}
The 'grouporder' of this group is the same as for order 4 thus 6 elements: N₉ ₃N₁₃N₉ N₃N₃ {₃N₁₃N₉}(₃N₁₃N₉) {₃N₁₃N₉}(N₃N₃) {N₃N₃}(₃N₁₃N₉)
Order 12 Dynamic numbering group
The vast order 12 DN-group with 41 generators shows elements that are not {panmagic} invariants most elements seem to be invariants, a subgroup might be definable that is {panmagic} invariant

Order 4 group theory
RLS_A abcd 0123 badc 1032 cdab 2301 dcba 3210	RLS_B abcd 0123 badc 1032 cdba 2310 dcab 3201	RLS_C abcd 0123 bcda 1230 cdab 2301 dabc 3012	RLS_D abcd 0123 bdac 1302 cadb 2031 dcba 3210
which expands to all 576 possibilities by performing 6 row permutations of type _2[0,perm(1,2,3)] and the 24 column permutations _1[perm(0,1,2,3)], further associating a,b,c,d with 0,1,2,3 gives the normally used form of latin squares on this site also shown above examination learns only RLS_A used in order 4 {panmagic) squares
RLS_A_{_2[0,2,1,3]} 0123 2301 1032 3210	RLS_A_{_2[0,1,3,2]=[0,3,1,2]} 0312 3021 2130 1203	RLS_A_{_2[0,3,1,2]} 0123 3210 1032 2301	RLS_A_{_2[0,2,3,1]=[0,3,1,2]} 0312 1203 2130 3021
Note that the latter 2 are related to the first 2 by the vertical "panflip" _2[0,3,2,1] while of course the first 2 are related by the combined _2[0,2,3,1]=[0,3,1,2] (!?) also see that RLS_A_{_2[0,3,1,2]} = RLS_A_{_2[0,2,3,1]=[0,3,1,2]}^t making the first a bit more natural then the latter since 4 * RLS_A_{_2[0,3,1,2]} + RLS_A_{_2[0,3,1,2]}^t is in it's normalized position immediately. As this LS has no equal elements symmetric to the maindiagonal it is orthogonal to it's transposed and thus all numbers are present in this square (the square thus obtained is #104 in my uploaded list which is but the vertical panflip of the first panmagic square (ie #104 = #102_{_2[0,3,2,1]}) which serves as entrypoint for the dynamic numbering-panflip group of grouporder 48 mentioned in the previous section)