Stabilizer in M_{24} of a dodecad

## 1. Mongean Shuffle

The Mongean shuffle concerns a deck of twelve cards, labeled 0 through 11. The permutation

r(t) = 11-t

reverses the cards around. The permutation

s(t) = min(2t,23-2t)

is called the **Mongean Shuffle**. The permutation group M_{12} is generated by r and s: M_{12} = < r,s >, as a subgroup of S_{12}. (See [12], Chap. 11, Sec. 17 or [18])

## 2. Steiner Hexad

Jacob Steiner (1796-1863) was a Swiss mathematician specializing in projective goemetry. (It is said that he did not learn to read or write until the age of 14 and only started attending school at the age of 18.) The origins of “Steiner systems” are rooted in problems of plane geometry. |

Let T be a given set with n elements. Then the **Steiner system **S(k,m,n) is a collection S = {S_{1}, … ,S_{r}} of subsets of T such that

- |S
_{i}| = m,
- For any subset R in T with |R| = k there is a unique i, 1<=i<=n such that R is contained in S
_{i}. |S(k,m,n)| = (n choose k)/(m choose k).

If any set H has cardinality 6 (respectively 8, 12) then H is called a **hexad**, (respectively **octad, dodecad**.)

Let’s look at the Steiner System S(5,6,12) and M_{12}. We want to construct the Steiner system S(5,6,12) using the projective line P^{1}(F_{11}). To define the hexads in the Steiner system, denote

- the projective line over F
_{11} by P^{1}(F_{11})={inf,0,1,…,10}.
- Q = {0,1,3,4,5,9}=the quadratic residues union 0
- G = PSL
_{2}(F_{11})
- S = set of all images of Q under G. (Each element g in G will send Q to a subset of P
^{1}(F_{11}). )

There are always six elements in such a hexad. There are 132 such hexads. If I know five of the elements in a hexad of S, then the sixth element is uniquely determined. Therefore S is a Steiner system of type (5,6,12).

**Theorem**:

M_{12} sends a hexad in a Steiner system to another hexad in a Steiner system. In fact, the automorphism group of a Steiner system of type (5,6,12) is isomorphic to M_{12}.

(For a proof, see [11], Theorem 9.78.)

The hexads of S form a Steiner system of type (5,6,12), so

M_{12} = < g in S_{12} | g(s) belongs to S, for all s in S > .

In other words, M_{12} is the subgroup stabilizing S. The hexads support the weight six words of the Golay code, defined next. (For a proof, see [6].)

## 3. Golay Code

” The Golay code is probably the most important of all codes for both practical and theoretical reasons.” ([17], pg. 64)
M. J. E. Golay (1902-1989) was a Swiss physicist known for his work in infrared spectroscopy among other things. He was one of the founding fathers of coding theory, discovering GC_{24} in 1949 and GC_{12} in 1954. |

A **code** C is a vector subspace of (F_{q})^{n }for some n >=1 and some prime power q =p^{k}.

An **automorphism** of C is a vector space isomorphism, f:C–>C.

If w is a code word in F_{q}^{n}, n>1, then the number of non-zero coordinates of w is called the **weight** w, denoted by wt(w). A **cyclic code** is a code which has the property that whenever (c_{0},c_{1},…,c_{n-1}) is a code word then so is (c_{n-1},c_{0},…,c_{n-2}).

If c=(c_{0},c_{1},…,c_{n-1}) is a code word in a cyclic code C then we can associate to it a polynomial g_c(x)=c_{0} + c_{1}x + … + c_{n-1}x^{n-1}. It turns out that there is a unique monic polynomial with coefficients in F_{q}

of degree >1 which divides all such polynomials g_c(x). This polynomial is called

a **generator polynomial** for C, denoted g(x).

Let n be a positive integer relatively prime to q and let *alpha* be a primitive n-th root of unity. Each generator g of a cyclic code C of length n has a factorization of the form *g(x) = (x-alpha*^{k1})… (x-alpha^{kr}), where {k_{1},…,k_{r}} are in {0,…,n-1} [17]. The numbers *alpha*^{k}_{i}, 1≤ i≤ r, are called the **zeros** of the code C.

If p and n are distinct primes and p is a square mod n, then the **quadratic residue code** of length n over F_{p} is the cyclic code whose generator polynomial has zeros

{alpha^{k} | k is a square mod n} [17]. The **ternary Golary code** GC_{11} is the quadratic

residue code of length 11 over F_{3}.

The **ternary Golay code GC**_{12 }is the quadratic residue code of length 12 over F_{3} obtained by appending onto GC_{11} a zero-sum check digit [12].

**Theorem**:

There is a normal subgroup N of Aut(GC_{12}) of order 2 such that Aut(GC_{12})/N is isomorphic to M_{12}. M_{12} is a quotient of Aut(GC_{12}) by a subgroup or order 2. In other words, M_{12} fits into the following short exact sequence:

1–>N–>Aut(GC_{12})–>M_{12}–>1

Where i is the embedding and N in Aut(GC_{12}) is a subgroup of order 2. See [6].

## 4. Hadamard Matrices

Jacques Hadamard (1865-1963) was a French mathematician who did important work in analytic number theory. He also wrote a popular book “*The psychology in invention in the mathematical field*” (1945). |

A **Hadamard matrix** is any n x n matrix with a +1 or -1 in every entry such that the absolute value of the determinant is equal to n^{n/2}.

An example of a Hadamard matrix is the Paley-Hadamard matrix. Let p be a prime of the form 4N-1, p > 3. A **Paley-Hadamard matrix** has order p+1 and has only +1’s and -1’s as entries. The columns and rows are indexed as (inf,0,1,2,…,p-1). The infinity row and the infinity column are all +1’s. The zero row is -1 at the 0th column and at the columns that are quadratic non-residues mod p; the zero row is +1 elsewhere. The remaining p-1 rows are cyclic shifts of the finite part of the second row. For further details, see for example [14].

When p = 11 this construction yields a 12×12 Hadamard matrix.

Given two Hadamard Matrices A, B we call them **left-equivalent** if there is an nxn signed permutation matrix P such that PA = B.

The set {P nxn signed permutation matrix| AP is left equivalent to A} is called the **automorphism group** of A. In other words, a matrix is an automorphism of the Hadamard matrix, if it is a nxn monomial matrix with entries in {0,+1,-1} and when it is multiplies the Hadamard matrix on the right, only the rows may be permuted, with a sign change in some rows allowed.

Two nxn Hadamard matrices A, B are called equivalent if there are nxn signed permutation matrices P_{1}, P_{2} such that A = P_{1} *B *P_{2}.

All 12×12 Hadamard matrices are equivalent ([13], [16] pg. 24). The group of automorphisms of any 12×12 Hadamard matrix is isomorphic to the Mathieu group M_{12 }([14] pg 99).

## 5. 5-Transitivity

Emile Mathieu (1835-1890) was a mathematical physicist known for his solution to the vibrations of an elliptical membrane. |

The fact that M_{12} acts 5-transitively on a set with 12 elements is due to E. Mathieu who proved the result in 1861. (Some history may be found in [15].)

There are only a finite number of types of 5-transitive groups, namely S_{n} (n>=5), A_{n} (n>=7), M_{12} and M_{24}. (For a proof, see [11])

Let G act on a set X via phi : G–>S_{X}. G is **k-transitive** if for each pair of ordered k-tuples (x_{1}, x_{2},…,x_{k}), (y_{1},y_{2},…,y_{k}), all x_{i} and y_{i} elements belonging to X, there exists a g in G such that y_{i} = phi(g)(x_{i}) for each i in {1,2,…,k}.

M_{12} can also be constructed as in Rotman [11], using transitive extensions, as follows (this construction appears to be due originally to Witt). Let f_{a,b,c,d}(x)=(ax+b)/(cx+d), let

M_{10} = < f_{a,b,c,d}, f_{a’,b’,c’,d’ }|ad-bc is in F_{q}^{x}, a’d’-b’c’ is not in F_{q}^{x} >,

q = 9.

pi = generator of F_{9}^{x}, so that F_{9}^{x} = < pi> = C_{8}.

Using Thm. 9.51 from Rotman, we can create a transitive extension of M_{10}. Let omega be a new symbol and define

M_{11} = < M_{10}, h| h = (inf, omega)(pi, pi^{2})(pi^{3},pi^{7}) (pi^{5},pi^{6})>.

Let P^{1}(F_{9}) = {inf, 0, 1, pi, pi^{2}, … , pi^{7}}. Then M_{11} is four transitive on Y_{0} = P^{1}(F_{9}) union {omega}, by Thm 9.51.

Again using Thm. 9.51, we can create a transitive extension of M_{11}. Let sigma be a new symbol and define

M_{12} = < M_{11}, k>, where k = (omega, sig)(pi,pi^{3}) (pi^{2},pi^{6})(pi^{5},pi^{7}). M_{12} is 5-transitive on Y_{1} = Y_{0} union {sig}, by Th. 9.51.

Now that we constructed a particular group that is 5-transitive on a particular set with 12 elements, what happens if we have a group that is isomorphic to that group? Is this new group also 5-transitive?

Let G be a subgroup of S_{12} isomorphic to the Mathieu group M_{12}. Such a group was constructed in Section 1.

**Lemma**: There is an action of G on the set {1,2,…,12} which is 5-transitive.

** proof:** Let Sig : G –> M_{12} be an isomorphism. Define g(i) = Sig(g)(i), where i = {1,2,…,12}, g is in G. This is an action since Sig is an isomorphism. Sig^{-1}(h)(i) = h(i) for all g in M_{12}, i in Y_{1}. Using some h in M_{12}, any i_{1},…,i_{5}

in Y_{1} can be sent to any j_{1},…,j_{5} in Y_{1}. That is, there exists an h in M_{12 }such that h(i_{k}) = j_{k}, k= 1,…,5 since M_{12} is 5-transitive. Therefore, Sig^{-1}(h)(i_{k}) = j_{k} = g(i_{k}). This action is 5-transitive. QED

In fact, the following uniqueness result holds.

**Theorem:** If G and G’ are subgroups of S_{12 }isomorphic to M_{12} then they are conjugate in S_{12}.

(This may be found in [7], pg 211.)

## 6. Presentations

The presentation of M_{12} will be shown later, but first I will define a presentation.

Let G = < x_{1},…,x_{n} | R_{1}(x_{1},…x_{n}) = 1, …, R_{m}(x_{1},…,x_{n}) = 1> be the smallest group generated by x_{1},…,x_{n} satisfying the relations R_{1},…R_{m}. Then we say G has **presentation** with generators x_{1},…,x_{n} and relations R_{1}(x_{1},…x_{n}) = 1, …, R_{m}(x_{1},…,x_{n}) = 1.

**Example:** Let a = (1,2,…,n), so a is an n-cycle. Let C_{n} be the cyclic group, C_{n} = < a > =

{1,a,…,a^{n-1}}. Then C_{n} has presentation < x | x^{n}=1 > = all words in x, where x satisfies x^{n}.=1 In fact, < x | x^{n} = 1 > is isomorphic to < a >. Indeed, the isomorphism

< x | x^{n} = 1 > –> < a > is denoted by x^{k} –> a^{k}, 0 <= k <= n-1. Two things are needed for a presentation:

- generators, in this case x, and
- relations, in this case x
^{n} = 1.

**Example**: Let G be a group generated by a,b with the following relations; a^{2} = 1, b^{2} = 1, (ab)^{2} = 1:

G = < a,b | a^{2} = 1, b^{2} = 1, (ab)^{2} = 1 > = {1,a,b,ab}.

This is a non-cyclic group of order 4.

Two presentations of M_{12} are as follows:

M_{12} = < A,B,C,D | A^{11} = B^{5} = C^{2} = D^{2} = (BC)^{2} = (BD)^{2} = (AC)^{3 }= (AD)^{3} = (DCB)^{2} = 1, A^{B} =A^{3} >

= < A,C,D | A^{11} = C^{2} = D^{2} = (AC)^{3} = (AD)^{3} = (CD)^{10} = 1, A^{2}(CD)^{2}A = (CD)^{2} >.

In the first presentation above, A^{B} = B^{-1}AB. These are found in [6] and Chap. 10 Sec. 1.6 [12].

## 7. Crossing the Rubicon

The Rubicon is the nick-name for the Rubik icosahedron, made by slicing the icosahedron in half for each pair of antipodal vertices. Each vertex can be rotated by 2*pi/5 radians, affecting the vertices in that half of the Rubicon, creating a shape with 12 vertices, and six slices. The Rubicon and M_{12} are closely related by specific moves on the Rubicon.

Let f_{1}, f_{2}, …,f_{12 }denote the basic moves of the Rubicon, or a 2*pi/5 radians turn of the sub-pentagon about each vertex. Then according to Conway,

M_{12} = < x*y^{-1} | x,y are elements of {f_{1}, f_{2}, …,f_{12} } >.

Actually, if a twist-untwist move, x*y^{-1}, as above, is called a **cross of the Rubicon**, then *M*_{12} is generated by the crosses of the Rubicon! ([1], Chap. 11 Sec. 19 of [12])

## 8. M_{12} and the Minimog

Using the Minimog and C_{4} (defined below), I want to construct the Golay code GC_{12}.

The **tetracode** C_{4} consists of 9 words over F_{3}:

0 000, 0 +++, 0 ---, where 0=0, +=1, and -=2 all mod 3.
+ 0+-, + +-0, + -0+,
- 0-+, - +0-, - -+0.

Each (a,b,c,d) in C_{4} defines a linear function f : F_{3} –> F_{3}, where f(x) = ax+b, f(0) = b, f(1) = f(+) = c, f(2) = f(-) = d, and a is the “slope” of f. This implies a + b = c (mod 3), b – a = d (mod 3).

**Minimog**: A 4×3 array whose rows are labeled 0,+,-, that construct the Golay code in such a way that both signed and unsigned hexads are easily recognized.

A **col** is a word of length 12, weight 3 with a “+” in all entries of any one column and a “0” everywhere else. A **tet** is a word of length 12, weight 4 who has “+” entries in a pattern such that the row names form a tetracode word, and “0” entires elsewhere. For example,

_________ _________
| |+ | | |+ |
| |+ | |+| + |
| |+ | | | +|
--------- ---------
"col" "tet"

The above “col” has “+” entries in all entries of column 2, and “0” entries elsewhere.

The above “tet” has a “+” entry in each column. The row names of each “+” entry are +, 1, +, – respectively. When put together, + 0+- is one of the nine tetracode words.

**Lemma**: Each word belongs to the ternary Golay Code GC_{12} if and only if

- sum of each column = -(sum row
_{0})
- row
_{+} – row_{ –} is one of the tetracode words.

This may be found in [4].

**Example:**

|+|+ + +| col sums: ---- row_{+} - row_{-}: --+0
|0|0 + -| row_{0} sum: + = -(sum of each col)
|+|+ 0 -|

How do I construct a Golay code word using cols and tets? By the Lemma above, there are four such combinations of cols and tets that are Golay code words. These are: col – col, col + tet, tet – tet, col + col – tet.

**Example:**

col-col col+tet tet-tet col+col-tet
| |+ -| | |+ + | |+|0 + +| | |- + +|
| |+ -| |+| - | |-| - | |-| 0 +|
| |+ -| | | + +| | | -| | | + 0|
? ? ? ? + 0 ? - - ? - + + 0 + -

“Odd-Man-Out”: The rows are labeled 0,+,-, resp.. If there is only one entry in a column then the label of the corresponding row is the **Odd Man Out**. (The name of the odd man out is that of the corresponding row.) If there is no entry or more than one entry in the column then the odd man out is denoted by “?”.

For example, in the arrays above, the Odd-Men-Out are written below the individual arrays.

For the Steiner system S(5,6,12), the minimog is labeled as such:

______________
|0 3 inf 2 |
|5 9 8 10 |
|4 1 6 7 |
--------------

The four combinations of cols and tets above that construct a Golay code word yield all signed hexads. From these signed hexads, if you ignore the sign, there are 132 hexads of the Steiner system S(5,6,12) using the (o, inf, 1) labeling discussed in Section 9 below. There are a total of 265 words of this form, but there are 729 Golay code words total. So, although the above combinations yield all signed hexads, they do not yield all hexads of the Golay code ([12] pg. 321).

The hexad for the tet-tet according to the S(5,6,12) Minimog above would be (0, inf, 2, 5, 8, 7).

The rules to obtain each hexad in this Steiner system is discussed in Section 9 below.

A Steiner system of type (5,6,12) and the Conway-Curtis notation can be obtained from the Minimog. S_{12} sends the 3×4 minimog array to another 3×4 array. The group M_{12} is a subgroup of S_{12} which sends the Minimog array to another array also yielding S(5,6,12) in Conway-Curtis notation.

## 9. Kitten

The kitten is also an interesting facet of the Minimog. Created by R.T. Curtis,

kittens come from the construction of the Miracle Octal Generator, or MOG, also created by R.T. Curtis. (A description of the MOG would be too far afield for this post, but further information on the MOG can be gotten from [3] or [6].)

Suppose we want to construct a Steiner system from the set T = {0, 1, …, 10, inf}.

The kitten places 0, 1, and inf at the corners of a triangle, and then creates a rotational symmetry of triples inside the triangle according to R(y) = 1/(1-y) (as in [2], section 3.1). A kitten looks like:

infinity
6
2 10
5 7 3
6 9 4 6
2 10 8 2 10
0 1
Curtis' kitten

where 0, 1, inf are the ** points at infinity**.

Another kitten, used to construct a Steiner system from the set T = {0, 1, …, 10, 11} is

6
9
10 8
7 2 5
9 4 11 9
10 8 3 10 8
1 0
Conway-Curtis' kitten

The corresponding minimog is

_________________________
| 6 | 3 | 0 | 9 |
|-----|-----|-----|-----|
| 5 | 2 | 7 | 10 |
|-----|-----|-----|-----|
| 4 | 1 | 8 | 11 |
|_____|_____|_____|_____|

(see Conway [3]).

The first kitten shown consists of the three points at 0, inf, 1 with an arrangement of points of the plane corresponding to each of them. This correspondence is:

6 |10 | 3 5 | 7 |3 5 | 7 | 3
2 | 7 | 4 6 | 9 |4 9 | 4 | 6
5 | 9 | 8 2 |10 |8 8 | 2 |10
inf-picture 0-picture 1-picture

A union of two perpendicular lines is called a **cross**. There are 18 crosses of the kitten:

___________________________________________
|* * * |* * * |* * * |* | * | * |
|* | * | * |* * * |* * * |* * * |
|* | * | * |* | * | * |
-----------------------------------------
_________________________________________
|* | * |* * |* |* * | * |
|* | * |* * | * * | * | * |
|* * * |* * * | * | * * |* * |* * * |
-----------------------------------------
_________________________________________
|* * | * | * * | * | * * |* * |
|* * |* * |* |* * | * * | * |
| * |* * | * * |* * |* |* * |
------------------------------------------

A **square** is a complement of a cross. The 18 squares of a kitten are:

___________________________________________
| | | | * * |* * |* * |
| * * |* * |* * | | | |
| * * |* * |* * |* * |* * |* * |
-----------------------------------------
_________________________________________
| * * |* * | * | * * | * |* * |
| * * |* * | * |* | * * |* * |
| | |* * |* | * | |
-----------------------------------------
_________________________________________
| * |* * |* |* * |* | * |
| * | * | * * | * |* |* * |
|* * | * |* | * | * * | * |
-----------------------------------------

The rules to obtain a hexad in the {0,1,inf} notation are the following:

- A union of parallel lines in any picture,
- {0, 1, inf} union any line,
- {Two points at infinity} union {square in a picture corresponding to omitted point at infinity},
- {One point at infinity} union {cross in the corresponding picture at infinity}.

(See [2])

M_{12} is isomorphic to the group of automorphisms of the Steiner system S(5,6,12) in the Conway-Curtis notation.

## 10. Mathematical Blackjack or Mathieu’s 21

Mathematical Blackjack is a card game where six cards from the group {0,1,…,11} are laid out face up on a table. The rules are:

- each player must swap a card with a card from the remaining six, that is lower than the card on the table;
- the first player to make the sum of all six cards less than 21 loses.

According to Conway and Ryba [8, section V, part (d)], the winning strategy of this game is to choose a move which leaves a Steiner hexad from S(5,6,12) in the shuffle

notation, whose sum is greater than or equal to 21, on the table.

The shuffle notation for the hexad, used in the Mathematical Blackjack game, is shown below (see also the description in the hexad/blackjack page):

8 |10 |3 5 |11 |3 5 |11 |3
9 |11 |4 2 | 4 |8 8 | 2 |4
5 | 2 |7 7 | 9 |10 9 |10 |7
0-picture 1-picture 6-picture

Riddle: Assuming the strategy, player A just made a winning hexad move that will force player B to make the sum under 21 on his next turn. Joe Smith walks up to player B and offers to shuffle all 12 cards while player A isn’t looking, for a fee. Player B grabs at his chance thinking that a random shuffle will let him back in the game. How is it that player B still loses?

Joe is actually working for Player A. Joe does not shuffle the cards randomly, but instead uses the M_{12} group generated by r, s (see section 1) to shuffle the cards. Since the M_{12} group preserves hexads, player A still has a winning game. (He and Joe split the money.)

## 11. Sporadic Groups

A **simple group** is a group with no normal subgroups except itself and {1}. Most simple groups are from a family such as PSL_{2}(F_{p}), p>3 or A_{n}, n >= 5. However there exist some simple groups outside of such well known families. These are called sporadic simple groups. M_{12} is a sporadic simple group of order 95,040. The only smaller sporadic group is M_{11} of order 7,920. (See [10] pg. 211)

## 12. Stabilizer in M_{24} of a dodecad.

M_{24} is a sporadic simple group of order 244,823,040 containing M_{12} as a subgroup. The Steiner system S(5,8,24) is a collection of 8 element subsets, called octads, from a 24 element set X, with the property that any five elements in X determine a unique octad in the system. There are (24 choose 5)/(8 choose 5) = 759 of these octads. M_{24} is the subgroup of S_{X} which sends the set of octads to itself. Two octads, O_{1}, O_{2}, intersect in a subset of X of order 0,2,4,6 or 8 [14]. If |O_{1} intersect O_{2}| = 2 then O_{1} + O_{2} is order 12. Such a subset of X is called a **dodecad**. M_{12} is isomorphic to

{g in M_{24} | g(O_{1} + O_{2}) = (O_{1} + O_{2})} = the stablizer of the dodecad O_{1} + O_{2}.

(See [6] for details)

Much more information can be received from the references below or from the hexad/blackjack page.

## References

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_{12} and the ‘Kitten’ ,” **Computational Group Theory**, Academic Press, London, 1984.
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Science, New York University, 1973.
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Links, London Mathematical Society, Cambridge University

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Typed into html by wdj, 4-18-97.

Corrections 4-27-2001.

Last updated 2018-06-10.

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