26

Jac k Palmer Sander s

T : 1YL •*• M .

s

i s t h e map define d by T (x, A- • • A x. ) = x ,, . A-•»A x

/

a R a (R) *

x

a 1 k a ( l ) a

(kX)1

For 3 e S ( j ) , 8(V), and M

D / T N

a r e s i m i l a r l y d e f i n e d . Given a e S (k)

p IV;

and 3 e S(j), we define a©£ e S(k+j) by

a8$(i) = a(i) , for 1 ^ i ^ k;

= $(i-k) + k, for k+1 _ i _ k+j .

We often use 1 to denote the identity map, the identity permutation,

or the identity homomorphism. I = I x««.x if

m

times.

3im = { (tw • **,t ) It. = 0 or t. = 1 for some i}.

1 m ' l l

We need convenient ways to define permutations. If (p ,•••,p ) = P is

1 2

a z-tuple of distinct integers in 4Z, and if {s ,#,,,s } = {p ,«",p },

then there is a unique permutation 3 e S(z) such that

3(P) = (Pg(1) '**"'p8(z)* = *sl'""'Sz* = S* ThUS the ecluation ^(p) = s

defines the element 3 e S(z). In general, however, we will be defining

permutations using the arbitrary k- and j-tuples R and V, whose entries

need notbe distinct. Therefore we adopt the convention that indefining a

permutation in this manner we always mean theunique permutation determined

when all entries aredistinct. For example, given a e S(k) and 3 £ S(j),

define y e S(k+j) by y(R,V) = (a(R),3(V)). Then y = a © 3.