Editing Conway group (section)

==Monomial subgroup N of Co<sub>0</sub>==
Conway started his investigation of Co<sub>0</sub> with a subgroup he called '''N''', a [[holomorph (mathematics)|holomorph]] of the (extended) [[binary Golay code#Mathematical definition|binary Golay code]] (as [[diagonal matrices]] with 1 or −1 as diagonal elements) by the [[Mathieu group M24|Mathieu group M<sub>24</sub>]] (as [[permutation matrices]]). {{nowrap|'''N''' ≈ 2<sup>12</sup>:M<sub>24</sub>}}.

A standard [[binary Golay code#A convenient representation|representation]], used throughout this article, of the binary Golay code arranges the 24 co-ordinates so that 6 consecutive blocks (tetrads) of 4 constitute a [[Mathieu group M24#Sextet subgroup|sextet]].

The matrices of Co<sub>0</sub> are [[orthogonal#Definitions|orthogonal]]; i. e., they leave the inner product invariant. The [[matrix (mathematics)#Invertible matrix and its inverse|inverse]] is the [[transpose]]. Co<sub>0</sub> has no matrices of [[matrix (mathematics)#Determinant|determinant]] −1.

The Leech lattice can easily be defined as the '''Z'''-[[module (mathematics)|module]] generated by the set Λ<sub>2</sub> of all vectors of type 2, consisting of
: (4, 4, 0<sup>22</sup>)
: (2<sup>8</sup>, 0<sup>16</sup>)
: (−3, 1<sup>23</sup>)

and their images under '''N'''. Λ<sub>2</sub> under '''N''' falls into 3 [[Group action (mathematics)#Orbits and stabilizers|orbit]]s of sizes [[Leech lattice#Geometry|1104, 97152, and 98304]]. Then {{nowrap|1={{abs|Λ<sub>2</sub>}} = {{val|fmt=commas|196560}} = 2<sup>4</sup>&sdot;3<sup>3</sup>&sdot;5&sdot;7&sdot;13}}. Conway strongly suspected that Co<sub>0</sub> was [[Group action (mathematics)#Remarkable properties of actions|transitive]] on Λ<sub>2</sub>, and indeed he found a new matrix, not [[generalized permutation matrix|monomial]] and not an integer matrix.

Let ''η'' be the 4-by-4 matrix
:<math>\frac{1}{2}\begin{pmatrix}
   1 &  -1 & -1 & -1 \\
  -1 &   1 & -1 & -1 \\
  -1 &  -1 &  1 & -1 \\
  -1 &  -1 & -1 &  1
\end{pmatrix}</math>

Now let ζ be a block sum of 6 matrices: odd numbers each of ''η'' and −''η''.<ref>Griess, p.&nbsp;97.</ref><ref>Thomas Thompson, pp.&nbsp;148–152.</ref> ''ζ'' is a [[symmetric matrix|symmetric]] and orthogonal matrix, thus an [[involution (mathematics)#group theory|involution]]. Some experimenting shows that it interchanges vectors between different orbits of '''N'''.

To compute |Co<sub>0</sub>| it is best to consider Λ<sub>4</sub>, the set of vectors of type 4. Any type 4 vector is one of exactly 48 type 4 vectors congruent to each other modulo 2Λ, falling into 24 orthogonal pairs {{nowrap|{''v'', –''v''}.}} A set of 48 such vectors is called a '''frame''' or '''cross'''. '''N''' has as an [[Group action (mathematics)#Orbits and stabilizers|orbit]] a standard frame of 48 vectors of form (±8, 0<sup>23</sup>). The subgroup fixing a given frame is a [[conjugacy class#Conjugacy of subgroups and general subsets|conjugate]] of '''N'''. The group 2<sup>12</sup>, isomorphic to the Golay code, acts as sign changes on vectors of the frame, while M<sub>24</sub> permutes the 24 pairs of the frame. Co<sub>0</sub> can be shown to be [[Group action (mathematics)#Remarkable properties of actions|transitive]] on Λ<sub>4</sub>. Conway multiplied the order 2<sup>12</sup>|M<sub>24</sub>| of '''N''' by the number of frames, the latter being equal to the quotient {{nowrap|1={{abs|Λ<sub>4</sub>}}/48 = {{val|fmt=commas|8,252,375}} = 3<sup>6</sup>&sdot;5<sup>3</sup>&sdot;7&sdot;13}}. That product is the order of ''any'' subgroup of Co<sub>0</sub> that properly contains '''N'''; hence '''N''' is a maximal subgroup of Co<sub>0</sub> and contains 2-Sylow subgroups of Co<sub>0</sub>. '''N''' also is the subgroup in Co<sub>0</sub> of all matrices with integer components.

Since Λ includes vectors of the shape {{nowrap|(±8, 0<sup>23</sup>)}}, Co<sub>0</sub> consists of rational matrices whose denominators are all divisors of 8.

The smallest non-trivial representation of Co<sub>0</sub> over any field is the 24-dimensional one coming from the Leech lattice, and this is faithful over fields of characteristic other than 2.