Dodecahedral Faces of M₁₂

by Ann Luers Casey

This post constitutes part of the math honors thesis written in spring 1997 at the USNA, advised by David Joyner. It is in the public domain.

Groups are objects in mathematics that measure symmetry in nature. A group is a set with a binary operation that has an inverse, an identity and is associative. For example, a clock has 12-fold symmetry. A more unusual group is a sporadic, non-abelian simple group. It can be very interesting to look more closely at such a group that arises naturally. One such group is M₁₂. This post explores two different ways of creating M₁₂ and then looks at twelve different ways M₁₂ appears in mathematics, hence the pun the “dodecahedral faces” in the title. Specifically, this post relates M₁₂ to the Mongean shuffle, hexads of a Steiner system, Golay codes, the Hadamard matrix of order 12, 5-transitivity, presentations, crossing the Rubicon, the minimog, the kitten, mathematical blackjack, sporadic groups, and the stabilizer in M₂₄ of a dodecad.

Definitions:

Homomorphism: Let G₁, G₂ be groups with *₁ denoting the group operation for G₁ and *₂ the group operation for G₂. A function f : G₁–>G₂ is a homomorphism if and only if for all a,b, in G₁ we have

f(a *₁ b) = f(a) *₂ f(b).

Isomorphism: If a homomorphism is bijective, then it is called an
isomorphism.

Automorphism: An isomorphism from a group G to itself is an automorphism.

Notation:

• Let F_q denote the finite field with q elements, q is a power of a prime.
• Z = the invertible scalar 2×2 matrices with entries in F_q^x.
• Let PGL₂(F_q) = GL₂(F_q)/Z = {A*Z | A is in GL₂(F_q)}, with multiplication given by
  (A*Z)(B*Z) = (A*B)Z. This is the projective linear group over F_q.
• LF(F_q) is the group of linear fractional transformations x–>(ax+b)/(cx+d).

Claim: There is a group theoretic isomorphism between PGL₂(F_q) and LF(F_q). (See [11], Theorem 9.47 for a proof.)

Claim: LF(F_q) acts 3-transitively on the set P¹(F_q) (q>3). I.e., one can send any triple to any other triple in P¹(F_q) by using a suitable linear fractional transformation. (See [11], Theorem 9.48 for a proof.)

Theorem

PSL₂(F_q) = < x–>x+1, x–>kx, x–>-1/x>, where k is any element in F_q^x that generates the multiplicative group of squares.

For a proof, see [12], ch 10, section 1.

One way to construct the Mathieu group M₁₂ is the following, accredited to Conway.

M₁₂ = < PSL₂(F₁₁), (2 10)(3 4)(5 9)(6 7) >.More explicitly, let

• f₁ be a cyclic permutation = x–> x+1 = (0,1,2,…,10)(inf).
• f₂ = x–>kx = (0)(1 3 9 5 4)(2 6 7 10 8)(inf) when k=3.
• f₃ = x–>-1/x = (0 inf)(1 10)(2 5)(3 7)(4 8)(6 9).
• f₄ = (2 10)(3 4)(5 9)(6 7).

Then M₁₂ = < f₁, f₂, f₃, f₄ >. Therefore, M₁₂ is a subgroup of the symmetric group on 12
symbols, namely inf, 0, 1, …, 10.

Another way to construct M₁₂ is given later under 5-transitivity.

There are many occurrences of M₁₂ in mathematics, but here I will list and explain twelve of them:

1. Mongean Shuffle
2. Steiner Hexad
3. Golay Code
4. Hadamard Matrices
5. 5- Transitivity
6. Presentations
7. Crossing the Rubicon
8. <a href="m_12.htm#M₁₂ and the Minimog”>M₁₂ and the Minimog
9. Kitten
10. Mathematical Blackjack or Mathieu’s 21
11. Sporadic Groups
12. <a href="m_12.htm#Stabilizer in M₂₄ of a dodecad”>Stabilizer in M₂₄ 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 M₁₂ is generated by r and s: M₁₂ = < r,s >, as a subgroup of S₁₂. (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 = {S₁, … ,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 M₁₂. We want to construct the Steiner system S(5,6,12) using the projective line P¹(F₁₁). To define the hexads in the Steiner system, denote

    • the projective line over F₁₁ by P¹(F₁₁)={inf,0,1,…,10}.
    • Q = {0,1,3,4,5,9}=the quadratic residues union 0
    • G = PSL₂(F₁₁)
    • S = set of all images of Q under G. (Each element g in G will send Q to a subset of P¹(F₁₁). )

    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:
    M₁₂ 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 M₁₂.

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

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

    M₁₂ = < g in S₁₂ | g(s) belongs to S, for all s in S > .

    In other words, M₁₂ 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 GC24 in 1949 and GC12 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_qⁿ, 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 (c₀,c₁,…,c_n-1) is a code word then so is (c_n-1,c₀,…,c_n-2).
    If c=(c₀,c₁,…,c_n-1) is a code word in a cyclic code C then we can associate to it a polynomial g_c(x)=c₀ + c₁x + … + c_n-1x^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^k_r), where {k₁,…,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 GC₁₁ is the quadratic
    residue code of length 11 over F₃.

    The ternary Golay code GC₁₂ is the quadratic residue code of length 12 over F₃ obtained by appending onto GC₁₁ a zero-sum check digit [12].

    Theorem:
    There is a normal subgroup N of Aut(GC₁₂) of order 2 such that Aut(GC₁₂)/N is isomorphic to M₁₂. M₁₂ is a quotient of Aut(GC₁₂) by a subgroup or order 2. In other words, M₁₂ fits into the following short exact sequence:

    1–>N–>Aut(GC₁₂)–>M₁₂–>1

    Where i is the embedding and N in Aut(GC₁₂) 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 P₁, P₂ such that A = P₁ *B *P₂.

    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 M₁₂ ([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 M₁₂ 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), M₁₂ and M₂₄. (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 (x₁, x₂,…,x_k), (y₁,y₂,…,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}.

    M₁₂ 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

    M₁₀ = < 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 F₉^x, so that F₉^x = < pi> = C₈.

    Using Thm. 9.51 from Rotman, we can create a transitive extension of M₁₀. Let omega be a new symbol and define

    M₁₁ = < M₁₀, h| h = (inf, omega)(pi, pi²)(pi³,pi⁷) (pi⁵,pi⁶)>.

    Let P¹(F₉) = {inf, 0, 1, pi, pi², … , pi⁷}. Then M₁₁ is four transitive on Y₀ = P¹(F₉) union {omega}, by Thm 9.51.

    Again using Thm. 9.51, we can create a transitive extension of M₁₁. Let sigma be a new symbol and define

    M₁₂ = < M₁₁, k>, where k = (omega, sig)(pi,pi³) (pi²,pi⁶)(pi⁵,pi⁷). M₁₂ is 5-transitive on Y₁ = Y₀ 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 S₁₂ isomorphic to the Mathieu group M₁₂. 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 –> M₁₂ 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 M₁₂, i in Y₁. Using some h in M₁₂, any i₁,…,i₅

    in Y₁ can be sent to any j₁,…,j₅ in Y₁. That is, there exists an h in M₁₂ such that h(i_k) = j_k, k= 1,…,5 since M₁₂ 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 S₁₂ isomorphic to M₁₂ then they are conjugate in S₁₂.

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

    6. Presentations

     

    The presentation of M₁₂ will be shown later, but first I will define a presentation.

    Let G = < x₁,…,x_n | R₁(x₁,…x_n) = 1, …, R_m(x₁,…,x_n) = 1> be the smallest group generated by x₁,…,x_n satisfying the relations R₁,…R_m. Then we say G has presentation with generators x₁,…,x_n and relations R₁(x₁,…x_n) = 1, …, R_m(x₁,…,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 | xⁿ=1 > = all words in x, where x satisfies xⁿ.=1 In fact, < x | xⁿ = 1 > is isomorphic to < a >. Indeed, the isomorphism
    < x | xⁿ = 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 xⁿ = 1.

    Example: Let G be a group generated by a,b with the following relations; a² = 1, b² = 1, (ab)² = 1:

    G = < a,b | a² = 1, b² = 1, (ab)² = 1 > = {1,a,b,ab}.

    This is a non-cyclic group of order 4.

    Two presentations of M₁₂ are as follows:

    M₁₂ = < A,B,C,D | A¹¹ = B⁵ = C² = D² = (BC)² = (BD)² = (AC)³ = (AD)³ = (DCB)² = 1, A^B =A³ >

    = < A,C,D | A¹¹ = C² = D² = (AC)³ = (AD)³ = (CD)¹⁰ = 1, A²(CD)²A = (CD)² >.

    In the first presentation above, A^B = B^-1AB. 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 M₁₂ are closely related by specific moves on the Rubicon.

    Let f₁, f₂, …,f₁₂ 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,

    M₁₂ = < x*y^-1 | x,y are elements of {f₁, f₂, …,f₁₂ } >.

    Actually, if a twist-untwist move, x*y^-1, as above, is called a cross of the Rubicon, then M₁₂ is generated by the crosses of the Rubicon! ([1], Chap. 11 Sec. 19 of [12])

    <a name="M₁₂ and the Minimog”>

    8. M₁₂ and the Minimog

     

    Using the Minimog and C₄ (defined below), I want to construct the Golay code GC₁₂.

    The tetracode C₄ consists of 9 words over F₃:

      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 C₄ defines a linear function f : F₃ –> F₃, 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 GC₁₂ if and only if

    • sum of each column = -(sum row₀)
    • row₊ – row _– is one of the tetracode words.

    This may be found in [4].

    Example:

    |+|+ + +|      col sums: ----      row+ - row-: --+0
    |0|0 + -|      row0 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. S₁₂ sends the 3×4 minimog array to another 3×4 array. The group M₁₂ is a subgroup of S₁₂ 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])

    M₁₂ 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 M₁₂ group generated by r, s (see section 1) to shuffle the cards. Since the M₁₂ 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 PSL₂(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. M₁₂ is a sporadic simple group of order 95,040. The only smaller sporadic group is M₁₁ of order 7,920. (See [10] pg. 211)

    <a name="Stabilizer in M₂₄ of a dodecad”>

    12. Stabilizer in M₂₄ of a dodecad.

     

    M₂₄ is a sporadic simple group of order 244,823,040 containing M₁₂ 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. M₂₄ is the subgroup of S_X which sends the set of octads to itself. Two octads, O₁, O₂, intersect in a subset of X of order 0,2,4,6 or 8 [14]. If |O₁ intersect O₂| = 2 then O₁ + O₂ is order 12. Such a subset of X is called a dodecad. M₁₂ is isomorphic to

    {g in M₂₄ | g(O₁ + O₂) = (O₁ + O₂)} = the stablizer of the dodecad O₁ + O₂.
    (See [6] for details)

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

    References

     

    1. W. D. Joyner, Mathematics of the Rubik’s Cube (USNA Course notes), 1997.
    2. R. T. Curtis, “The Steiner System S(5,6,12), the Mathieu Group M₁₂ and the ‘Kitten’ ,” Computational Group Theory, Academic Press, London, 1984.
    3. J. H. Conway, “hexacode and Tetracode- MOG and MINIMOG,” Computational Group Theory (ed. Atkinson), Academic Press, London, 1984.
    4. Vera Pless, “Decoding the Golay Code,” Transactions of Information Theory, IEEE, 1986, (pgs 561-567).
    5. R. T. Curtis, “A new Combinatorial approach to M₂₄“, Mathematical Proceeding of the Cambridge Philosophical Society, Vol. 79, 1974.
    6. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, “M₁₂,”,
       Atlas of Finite Groups, Clarendon Press, Oxford, 1985.
    7. Robinson, A Course in the Theory of Groups, Springer, 1996.
    8. J. H. Conway, N. Sloane, “Lexicographic Codes: Error-Correcting Codes
       from Game Theory,” Transactions on Information Theory, IEEE, 1986.
    9. A .Adler, “The modular Curve X(11) and the Mathieu group M₁₁“,
       Proc. London Math Society 74(1997)1-28.
       See also the paper X(11) and M₁₁.
    10. T. Thompson, From Error-Correcting Codes Through Sphere
        Packings to Simple Groups, The Mathematical Association of
        America, 1983.
    11. Rotman, J, Introduction to the Theory of Groups, 4th ed.
        Springer-Verlag, 1995.
    12. J. Conway, N. Sloane, Sphere Packings, Lattices, and Groups,
        Springer-Verlag, 3rd ed., 1999.
    13. B. Kostant, “The Graph of the truncated icosahedron and the
        last letter of Galois.” Notices of the A.M.S. 42(1995)959-
        968.
    14. E. Assmus, “On the Automorphism Groups of Paley-Hadamard
        Matrices.” Combinatorial Mathematics and its Applications.
        University of North Carolina Press, 1969, (pgs 98-103).
    15. P. Greenberg, Mathieu Groups, Courant Institute of Math and
        Science, New York University, 1973.
    16. P. Cameron, J. Van Lint, Designs, Graphs, Codes, and Their
        Links, London Mathematical Society, Cambridge University
        Press, 1991.
    17. F. MacWilliams, N. Sloane, The Theory of Error Correcting
        Codes, North Holland Publishing Company, 1978.
    18. R. Graham, P. Diaconis, W. Kantor, “The Mathematics of
        Perfect Shuffles”, Advanced Applied Math, Vol. 4, 1985, (pgs
        175-196).

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    Corrections 4-27-2001.
    Last updated 2018-06-10.