# Does trans-decalin really have a plane of symmetry?

My teacher told me that trans-decalin (see below) is achiral due to the presence of both, a centre of symmetry and a plane of symmetry. But I could not spot the plane of symmetry untill now. Can someone point out the plane of symmetry in a diagram?

• It passes through the two central carbons, perpendicular to the molecule. – Ivan Neretin Oct 13 '16 at 8:22

The question has already been answered nicely by Martin - マーチン♦ who pointed out that there is a mirror plane (i.e. a $\sigma = S_1$ element) and a center of inversion (i.e. an $S_2$ element), which both are already enough for a molecule to become achiral (existence of any $S_n$ element suffices for that). There is one important caveat though that often bugged me during stereochemistry lectures: Even if the ground state of a substance might not be achiral (for example if you placed some substituents on your decalin), then it can still be possible for the chemical to be achiral due to quick racemization when an achiral transition state with low energy is available (e.g. inversion of $\mathrm{NR_3},\mathrm{PR_3}$ systems or the $^1C_4\mapsto ^4C_1$ inversion of cyclohexanes). I think that is a little trap that is often missed in stereochemistry lectures.