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Let G be a finite group written additively and S = (G, G, δ) a group semiautomaton. In this paper we investigate the subnearring N (σ, τ) (S) of M(G), referred to as the syntactic nearring of S, with δ(x,y) = xσ + yτ for σ, τ, ∈ M(G). By definition, N(σ, τ)(S) is the subnearring of M(G) generated by under pointwise addition and composition of mappings. We first prove the key result that if (G,+) is a finite group, σ ∈ M(G) and m, n ∈ Z, then. Finally, we apply the result to the case where G = S3, the symmetric group of degree 3, and give a detailed study for the case S = (S3, S3, δ) with δ(x,y) = mx + ny for various syntactic nearrings.
KeywordsNormal Subgroup Transition Function Finite Group Symmetric Group Dihedral Group
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