Decision Problems in Restricted Classes of Groups

139 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1980.It is known that many decision problems are unsolvable in the class of all finitely presented groups. When the class of groups is restricted, problems previously unsolvable can become solvable. In this work we investigate decision problems for various restricted classes of groups.We first consider the classes of finitely presented groups for which each group has solvable (lamda)-problem, where (lamda) is conjugacy, order and power. We show that "Markov-type" problems are unsolvable in these classes. We next consider recursively enumerable classes of group presentations (both finite presentations and infinite presentations) that have uniformly solvable word problem. We show that for classes of recursively generated, recursively related presentations of this type most problems are unsolvable. For classes of finite presentations with uniformly solvable word problem, the properties "having solvable order problem", "having solvable power problem", and "being torsion free" are recursively unrecognizable. Finally, we investigate the triviality problem for classes of finite presentations of groups and semigroups with deficiency close to zero. We show that the class of semigroup presentations of deficiency -1 and the class of group presentations of deficiency -11 have unsolvable triviality problem.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD