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1,2,3-Trimethylcyclopropane has two stereoisomers. When one of the methyl groups is above the plane and other two are below the plane, then there are two chiral centres as well as a plane of symmetry. By definition this compound should have a meso isomer. However I read that this compound has zero meso isomers since it does not have any optically active stereoisomers. So what is the exact definition of meso?

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The way that a meso compound is often described in introductory organic chemistry courses is a compound that has stereocenters but is not chiral (due to a symmetry element). This description works for most cases, such as those in the previous questions (see this or this). By this description, the stereoisomers of 1,2,3-trimethylcyclopropane are meso.

However, the IUPAC Gold Book definition of a meso compound is:

A term for the achiral member(s) of a set of diastereoisomers which also includes one or more chiral members.

As is pointed out in the question, none of the diastereoisomers of 1,2,3-trimethylcyclopropane are chiral, so by this definition, none of these compounds are meso.

The Gold Book also has a description of the meso prefix:

A prefix to indicate the presence of a symmetry element of the second kind (see chirality) in a chemical species which is usually one of a set of diastereoisomers that also includes a chiral member.

The usage of meso in this way, to make a particular call out for a member in a stereoisomeric set of compounds, fails for 1,2,3-trimethylcyclopropane. Because both of the stereoisomers are achiral, neither can unambiguously be called "meso-1,2,3-trimethylcyclopropane." Since that's the case, neither are meso.

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    $\begingroup$ Just as an aside, I like the more common description (has stereocenters, but achiral) better, because I think that's an interesting situation that deserves a name. Congratulations on finding a unique case that forces us to think about the precise definition of these terms. $\endgroup$ – jerepierre Jun 11 '16 at 0:08

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