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Given a permutation automaton (or pure-group automaton), its transition monoid is a finite permutation group K and its language is a group language (say of a group G). Is it known whether there is a connection between K and the properties of the group G? More precisely, given K, can we say something on the group G? I would be happy to get some reference on that topic.

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    I would ask that on the professional maths site http://mathoverflow.net/. Or the CS site. – 2017-02-15
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    What do you mean by "is a group language, say of a group $G$"? Do you mean that the syntactic monoid of the language is $G$ or only that the language is recognised by $G$? – 2017-02-21
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    @mwm It would be a very bad idea to ask this question on mathoverflow, as it is not a research level question. Furthermore, one should not post the same question to different sites (unless it is unanswered for a long time). – 2017-02-21
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    I mean that the language is recognized by G – 2017-02-22

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Let $L$ be a regular language of $A^*$ and let $\mathcal{A}$ be its minimal automaton. Then the transition monoid $K$ of $\mathcal{A}$ is the syntactic monoid of $L$. If $\mathcal{A}$ is a permutation automaton, then $K$ is a permutation group.

Now, if a finite group $G$ recognizes $L$, then there exists by definition a monoid morphism $f:A^* \to G$ and a subset $P$ of $G$ such that $L = f^{-1}(P)$. Then $H = f(A^*)$ is as subgroup of $G$ (of course $G = H$ if you assume $f$ to be surjective) and $K$ is a quotient of the group $H$. Possible reference: Chapter 4, Section 4 of this link.

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    I will look at the reference, it seems very complete. Thanks a lot! – 2017-02-28