Number Theory: An Introduction to Mathematics

7 Groups and Codes 251

λ′=λ(v−t+ 1 )/(k−t+ 1 )

does not depend on the choice ofSand at-(v,k,λ)design (P,B)isalsoa(t− 1 )-

(v,k,λ′)design. By repeating this argument,we see that each point is contained in

exactlyrblocks, where

r=λ(v−t+ 1 )···(v− 1 )/(k−t+ 1 )···(k− 1 ),

and the total number of blocks isb=rv/k. In particular, anyt-design witht>2is
also a 2-design.
With any Hadamard matrixA =(αjk)of ordern= 4 dthere is, in addition,
associated a 3-( 4 d, 2 d,d− 1 )design. For assume without loss of generality that all
elements in the first column ofAare 1. We takeP={ 1 , 2 ,...,n}as the set of points
and{B 2 ,...,Bn,B 2 ′,...,Bn′}as the set of blocks, whereBk ={j∈P:αjk= 1 }
andBk′ ={j∈ P:αjk=− 1 }. Then, by Proposition 5,|Bk|=|Bk′|=n/2for
k= 2 ,...,n.IfTis any subset ofPwith|T|=3, sayT={i,j,}, then the number
of blocks containingTis the number ofk>1 such thatαik=αjk=αk.But,by
Proposition 5 again, the number of columns ofAwhich have the same entries in rows
i,j,isn/4 and this includes the first column. HenceTis contained in exactlyn/ 4 − 1
blocks. Again the argument may be reversed to show that any 3-( 4 d, 2 d,d− 1 )design
is associated in this way witha Hadamard matrix of order 4d.

7 Groups and Codes

A group is said to besimpleif it contains more than one element and has nonor-

malsubgroups besides itself and the subgroupcontaining only the identity element.

The finite simple groups are in some sense the building blocks from which all finite

groups are constructed. There are several infinite families of them: the cyclic groups

Cpof prime orderp, the alternating groupsAnof all even permutations ofnobjects

(n≥ 5 ), the groupsPSLn(q)derived from the general linear groups of all invertible

linear transformations of ann-dimensional vector space over a finite field ofq=pm

elements(n≥2andq>3ifn= 2 ), and some other families similar to the last which

are analogues for a finite field of the simple Lie groups.

In addition to these infinite families there are 26sporadicfinite simple groups.

(Theclassification theoremstates that there are no other finite simple groups besides

those already mentioned. The proof of the classification theorem at present occupies

thousands of pages, scattered over a variety of journals, and some parts are actually

still unpublished.) All except five of the sporadic groups were found in the years

1965–1981. However, the first five were found by Mathieu (1861,1873):M 12 is a

5-fold transitive group of permutations of 12 objects of order 12· 11 · 10 · 9 ·8and

M 11 the subgroup of all permutations inM 12 which fix one of the objects;M 24 is a

5-fold transitive group of permutations of 24 objects of order 24· 23 · 22 · 21 · 20 ·48,

M 23 the subgroup of all permutations inM 24 which fix one of the objects andM 22 the

subgroup of all permutations which fix two of the objects. The Mathieu groups may

be defined in several ways, but the definitions by means of Hadamard matrices that we

are going to give are certainly competitive with the others.