Point Reflection Group
The composition of two point reflections is a translation. Specifically, point reflection at p followed by point reflection at q is translation by the vector 2(q – p).
The set consisting of all point reflections and translations is Lie subgroup of the Euclidean group. It is a semidirect product of Rn with a cyclic group of order 2, the latter acting on Rn by negation. It is precisely the subgroup of the Euclidean group that fixes the line at infinity pointwise.
In the case n = 1, the point reflection group is the full isometry group of the line.
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