Finitely presented and conjugacy-separable implies solvable conjugacy problem

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., finitely presented conjugacy-separable group) must also satisfy the second group property (i.e., group with solvable conjugacy problem)
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Statement

A finitely presented conjugacy-separable group (i.e., a finitely presented group that is also a conjugacy-separable group) is a group with solvable conjugacy problem.

Related facts

Similar facts

• Finitely presented and residually finite implies solvable word problem

Facts used

1. Finitely presented implies all homomorphisms to any finite group can be listed in finite time
2. Conjugacy-separable and all homomorphisms to any finite group can be listed in finite time implies solvable conjugacy problem

Proof

The proof follows by combining Facts (1) and (2).