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.
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