By H. S Hall
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Therefore, A x A' holds and also A' is strongly connected. 6, G(A) M G(A') holds. This means that A' is regular. 1 Let A = ( S , X , S ) be a perfect automaton and let G be a group such that G M G(A). Then A is isomorphic to some (1,G)-automaton. 3. 5) Let A = ( S ,X, 6 ) be a strongly connected automaton. T h e n if IS1 = IG(A)I, A i s a permutation automaton. Proof By JSJ= JG(A)J,A is isomorphic to some (1,G)-automaton. 2, A is a permutation automaton. 1 Let A = X , 6,) be a regular ( n ,G)-automaton.
O)@(Z)= ( 0 , . . , O , O t ( g ' ) , 0 , . . , 0) and O ( ( g g ' , 0 , . . , O ) ) = O ( ( e ,0 , . . , O)YZ)= O ( ( e ,0 , . . , O ) ) @ ( Y Z=) (0,. . ,0, ht,O,. . ,O)@(YZ)= (0,. . ,O,Ot(gg'), 0 , . . , O ) . Hence the ( t ,t)entries of @ ( Y )@(Z) , and @(YZ) are h F I O t ( g ) ,h r l O t ( g ' ) and hFIOt (gg'), respectively. Notice that @ ( Y Z )= @(Y)@(Z). Consequently, hF1@t(gg') = h ; ' O t ( g ) h F 1 O t ( g ' ) and cp(gg') = cp(g)'p(g'). h) (-+) Assume that there exist cp, ki, i = 1,2,, .
N. By the fact that E is a regular system in G,, we can see that for any g E G there exist some group-matrices (ypq), ($,) E E" such that yT(+(i) = g and yb(i)T(j) = e. Thus we have @i(g)h,lhj = @ j ( g ) for any g E G and i , j = 1 , 2 , . . ,n. , 0)) = (0, . . , o ) , t = 1 , 2, . . ,n. P u t cp(g) = h ; l @ d g ) , g E G and ki = hF1hi E G , i = 1 , 2 , . . , n. Then we have h ; ' O j ( g ) = h;l&(g)h;'hj = k i l @ ( g ) k j for any g E Go a n d i , j = 1 , 2 , . . ,n. First we prove that cp is an automorphism of G.
A Short Introduction to Graphical Algebra by H. S Hall