Finiteness Conditions on Subgroups and Formal Language Theory

We show in this article that the most usual finiteness conditions on a subgroup of a finitely generated group all have equivalent formulations in terms of formal language theory. This correspondence gives simple proofs of various theorems concerning intersections of subgroups and the preservation of finiteness conditions in a uniform manner. We then establish easily the theorems of Greibach and of Griffiths by considering free reductions of languages that describe the computations of pushdown automata in one case and of Turing machines in the other, thus making clear that they are essentially the same.

[1]  A. V. Anisimov,et al.  Some algorithmic problems for groups and context-free languages , 1972 .

[2]  Richard Stanley.  Review: Sheila A. Greibach, A Note on Pushdown Store Automata and Regular Systems , 1968 .

[3]  Franz D. Seifert,et al.  Zur algebraischen Charakteristik der durch kontext-freie Sprachen definierten Gruppen , 1975, J. Inf. Process. Cybern..

[4]  A. G. Howson.  On the Intersection of Finitely Generated Free Groups , 1954 .

[5]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[6]  Note on a theorem of Schreier , 1957 .

[7]  G. Higman.  Subgroups of finitely presented groups , 1961, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[8]  Timothy V. Griffiths.  Some Remarks on Derivations in General Rewriting Systems , 1968, Inf. Control..

[9]  Donald F. Stanat,et al.  A Homomorphism Theorem for Weighted Context-Free Grammars , 1972, J. Comput. Syst. Sci..

[10]  守屋 悦朗,et al.  J.E.Hopcroft, J.D. Ullman 著, "Introduction to Automata Theory, Languages, and Computation", Addison-Wesley, A5変形版, X+418, \6,670, 1979 , 1980 .

[11]  G. Higman,et al.  Embedding Theorems for Groups , 1949 .

[12]  Leon Greenberg,et al.  Discrete Groups of Motions , 1960, Canadian Journal of Mathematics.

[13]  W. Magnus,et al.  Combinatorial Group Theory: COMBINATORIAL GROUP THEORY , 1967 .

[14]  Sheila A. Greibach,et al.  A note on pushdown store automata and regular systems , 1967 .