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An example of a finitely presented amenable group that does not belong to the class EG. Mat. , 189(1):79–100, 1998. S. Guba. On the properties of the Cayley graph of Richard Thompson’s group F . Internat. J. , 14(5-6):677–702, 2004. International Conference on Semigroups and Groups in honor of the 65th birthday of Prof. John Rhodes. S. V. Sapir. The Dehn function and a regular set of normal forms for R. Thompson’s group F . J. Austral. Math. Soc. Ser. A, 62(3):315–328, 1997. [14] Victor Guba and Mark Sapir.

Then τH is a cellular automaton over H with memory set S and local defining map µ. The cellular automaton τH is called the restriction of τ to H ([CeC1]). We shall use the following result whose proof is given in [CeC2]. 1. The cellular automaton τ is injective (resp. surjective) if and only if its restriction τH is injective (resp. surjective). Moreover, if A is a left module over a ring R, then τ is R-linear if and only if τH is R-linear. We shall also use the following. 2 (Closure Lemma). Let G be a countable group and let M be an Artinian left module over a ring R.

By the solution to the restricted Burnside problem, their group is not residually finite, since it contains an infinite, finitely generated, finite exponent group. – If we also drop the finite presentation assumption, there are many examples. One of the simplest (but not the best known) is given by the standard wreath product F Z, where F is any finite, non-solvable group. 2 Indeed, if k ≥ 4m − 1, then the elements b1 = 1, b2 = 3, . . 3) of [NN59]. Some Residually Finite Groups Satisfying Laws 49 – Again using the solution to the restricted Burnside problem, the residually finite groups whose derived subgroup has finite exponent are (locally finite)by-abelian.

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Abstract Algebra - A Comprehensive Trtmt by C. Menini, F. van Oystaeyen

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