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In this study, we introduce the natural mate and conjugate mate of a Frenetcurve in a three dimensional Lie group $ \mathbb{G} $ with bi-invariant metric.Also, we give some relationships between a Frenet curve and its natural mate orits conjugate mate in $ \mathbb{G} $. Especially, we obtain some results forthe natural mate and the conjugate mate of a Frenet curve in $ \mathbb{G} $when the Frenet curve is a general helix, a slant helix, a spherical curve, arectifying curve, a Salkowski (constant curvature and non-constant torsion),anti-Salkowski (non-constant curvature and constant torsion), Bertrand curve.Finally, we give nice graphics with numeric solution in Euclidean 3-space as acommutative Lie group.

Natural and Conjugate Mates of Frenet Curves in Three-Dimensional Lie Group