The Word Problem in Quotients of a Group
نویسنده
چکیده
Every finitely presented group G has a quotient group with solvable word problem – namely the trivial group. We will construct a finitely presented group G such that every non-trivial quotient of G has unsolvable word problem (of degree 0′ in case the quotient is recursively presented) and such that every countable group is embedded in some quotient of G. A presentation of a group is an ordered pair 〈S;D〉 where S is a set and D a collection of words on the elements of S and their inverses. The group G presented by 〈S;D〉 is the quotient of the free group on S by the normal closure of the words in D, written G = 〈S;D〉. We usually will not distinguish between a group (as an abstract algebraic object) and its presentation. G = 〈S;D〉 is said to be finitely generated if S is finite and finitely presented when S and D are both finite. In case S is finite and D is a recursively enumerable (r.e.) set of words, G is said to be recursively presented. (See Rogers [5] for all logical terminology.) In case G is finitely generated, the word problem for G is the algorithmic problem of deciding for arbitrary words w of G whether or not w = 1 in G. Suppose that G = 〈S;D〉 is finitely generated and let F denote the free group on S, K the normal closure of the words in D so that G = F/K. Let M be a normal subgroup of G (written M / G), NM the preimage of M in F . Then K ⊂ NM / F and G/M ∼= F/NM . The normal subgroup M of G is said to be r.e. if and only if NM is an r.e. set of words in F . Note that NM is an r.e. set of words if and only if NM is the normal closure in F or an r.e. set of words. Since G is finitely generated, M is r.e. if and only if G/M can be recursively presented. Finally, we remark that the degree of the word problem for G = 〈S;D〉 is just the degree of K as a subset of the words of F . For G recursively presented, the degree of the word problem is an r.e. degree. Also observe that G has solvable word problem if and only if K and F/K are both r.e. sets of words.
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تاریخ انتشار 2002