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Here, h is an arbitrary element in H . 6 Direct product of automata In this section, we will deal with direct products of group-matrix type automata and their automorphism groups. 1 Let A = (S,X , 6 ) and B = (T,X , y) be two automata. The direct product, A x B,is the automaton A x B = ( S x T , X , 6 x y) where 6 x y ( ( s , t ) , a ) = ( S ( s , a ) , y ( t , a ) )for any ( s , t ) E S x T and a E X . Before introducing the notion of the direct product of groupmatrix type automata, we define the product of two group-matrices.

3 . j called the characteristic monoid of A . 1 Let A = ( S , X , 6 ) and B = ( T , X , y ) be two isomorphic automata. T h e n C ( A )= C ( B ) . Then for any x,y E X* and s E S , S ( s , x ) = S(s,y) if and only if y(p(s),x) = y ( p ( s ) , y). Since p is a bijective mapping, this means that, for any s E S , 6 ( s ,x) = 6(s, y) if and only if ~ ( sz) , = y(s, y). Hence C ( A )= C ( B ) . (G,X , 6,p) be a n (n,G)-automaton. T h e n the characteristic monoid of A is isomorphic to Q f ( X * ) . 2 Let A = Proof Obvious from the fact that, for any x,y E only if @(z)= @(y).

Then fij = x g i k h k j fzT'. n # 0 and k=l h r j # 0 , we have Hi C_ girHrg,' and Hr & hrjHjhG1. e. Hi C_ f i j H j f i l . This completes the proof of (gpqHq)(hpqHq) E G(H(,)). such that fij = C g i k h f i j = g i r h r j . 4 Let H ( n ) = { H I ,Hz, H3,. . , H,} be a family of subgroups of a given finite group G satisfying the SP-condition. Then a vector ( f p H p )is called a generalized group-vector of order n over ( H ( n ) , G )and denoted ( f p H p )E 6 ( H ( n ) )if (fp) E G^, holds.