Search results
Results from the WOW.Com Content Network
In less formal terms, the group consists of words in the generators and their inverses, subject only to canceling a generator with an adjacent occurrence of its inverse. If G is any group, and S is a generating subset of G, then every element of G is also of the above form; but in general, these products will not uniquely describe an element of G.
The word problem asks whether two words are effectively the same group element. By relating the problem to Turing machines, one can show that there is in general no algorithm solving this task. Another, generally harder, algorithmically insoluble problem is the group isomorphism problem, which asks whether two groups given by different
In abstract algebra, the group isomorphism problem is the decision problem of determining whether two given finite group presentations refer to isomorphic groups.. The isomorphism problem was formulated by Max Dehn, [1] and together with the word problem and conjugacy problem, is one of three fundamental decision problems in group theory he identified in 1911. [2]
On the other hand, the fact that a particular algorithm does not solve the word problem for a particular group does not show that the group has an unsolvable word problem. For instance Dehn's algorithm does not solve the word problem for the fundamental group of the torus. However this group is the direct product of two infinite cyclic groups ...
The problem asks, for a particular class of finitely presented groups if there exists an algorithm that, given a finite presentation of a group from the class, computes the rank of that group. The rank problem is one of the harder algorithmic problems studied in group theory and relatively little is known about it. Known results include:
Let Q be a loop with abelian inner mapping group. Is Q nilpotent? If so, is there a bound on the nilpotency class of Q? In particular, can the nilpotency class of Q be higher than 3? Proposed: at Loops '07, Prague 2007; Comments: When the inner mapping group Inn(Q) is finite and abelian, then Q is nilpotent (Niemenaa and Kepka). The first ...
A very closely related topic is geometric group theory, which today largely subsumes combinatorial group theory, using techniques from outside combinatorics besides. It also comprises a number of algorithmically insoluble problems, most notably the word problem for groups; and the classical Burnside problem.
Pages in category "Group problem solving methods" The following 14 pages are in this category, out of 14 total. This list may not reflect recent changes. 0–9.