Download PDF by R. Keith Dennis: Algebraic K-Theory: Proceedings, Oberwolfach, FRG 1980

By R. Keith Dennis

ISBN-10: 0387119663

ISBN-13: 9780387119663

ISBN-10: 3540119663

ISBN-13: 9783540119661

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Extra resources for Algebraic K-Theory: Proceedings, Oberwolfach, FRG 1980

Example text

The following theorem plays a fundamental role in this section. 7. 1 Let A = X , 6,) be a regular (n,G)-automaton and E be a homomorphism of G onto some group. e. Ker(t) = J - l ( ( ( e ) )(e is the identity of G). 1 In the above, we identify G ( A )with G. 1). This assumption will be taken throughout this section. 1 X , G). Then for f^ the following: (z, - Put H = K e r ( J ) & G and A / H = = (fp) E G,, k = ( k p ) E G^, we have h - - f^ = holds if and only if there exists an element E h E H such that f = h($) = p h ( i ) = ( h k p ) .

We will obtain a representation of A by a group-matrix type automaton of order ISl/lG(A)I on G(A). e. P(P> = 4 , p ( q ) = P, p ( t ) = r and p(r) = t. ISI/IG(A)( = 2, x = { a , b , c ) , s = { P , q , r , t ) , s1 = b , q ) , s2 = { t , r ) , T = { p , t ) , J(p,a) = q = p ( p ) , J ( t , a ) = t = e ( t ) , +(a) = t = e(t),b(t,b) = p = e(p), Q(b) = 6(t,c)= r = p ( t ) , ~ ( c=) Then A' = ( G @ ) 2 , X , 6 q ) M A. 22 CHAPTER 1. 4 Equivalence of regular systems In this section, we will deal with the problem as noted in the previous page.

2 Let G be a finite group and let n be a positive integer. T h e n [ I ( G ) / n ] J ( n , G ) 5 [ I ( G ) / n l + p ( n ) where p(1) = 0 and p ( n ) = 1 f o r n 2 2. < Proof Obviously, the theorem holds true for the case n = 1. Therefore, we consider the case n >_ 2. 1. Hence we have to prove the inequality J ( n ,G ) 5 [ I ( G ) / n l P(4. e. [HI = G such that H = {hi I hi E G , i = 1 , 2 , . . , I ( G ) } . Now, put X = Y U { z } where Y = {yi I i = 1 , 2 , .. Moreover, for any i = 1 , 2 , .

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Algebraic K-Theory: Proceedings, Oberwolfach, FRG 1980 by R. Keith Dennis


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