The conjugacy problem for a group G is the problem of determining, given x, y ∈ G, whether or not there exists an element z ∈ G such that z-1xz = y. As one of the main decision problems in combinatorial group theory, there are many open questions which deal with the conjugacy problem.;This thesis discloses a new cyptosystem based on the conjugacy problem in groups and proves the following results (1) There exist a finitely presented group which has solvable word problem, unsolvable conjugacy problem and is right-orderable. (2) Every torsion-free group with solvable power problem be embedded in a group with solvable conjugacy problem. (3) The class of locally finite-indicable groups is not equal to the class of groups which have a normal system with finite factors. Equality does not even hold for the finitely generated case.;The cryptosystem is based on the braid group cryptosystem [15] but employs groups where the conjugacy problem is unsolvable. It may be possible to use the first result to prove an affirmative answer to the corresponding open question for lattice-orderability. The second result is already known, but the authors of the original proof state the following.;"The construction in the proof of this theorem is complicated and employs ideas of three previous papers..."[25].;In the present thesis we give a short, self-contained, and more direct proof which draws on only one previous result of [14] which is well known. The third result is the only section of this thesis which does not deal with the conjugacy problem for groups.
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