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http://en.wikipedia.org/wiki/Crystal_system

Now I forgot the bit that I've understood but before trying to study the topic again I would like to ask an other simple question: " What makes the 32 crystallographic point groups "different"? " (the question applies also to space groups).

I've found only just the enumeration of these groups and some kind of construction. But from the mathematical point of view I think that there are isomorphic point groups among these 32 (it can also be confirmed from wikipedia page, under the voice "abstract group" on the table of point gropus). Therefore group isomorphism it seems to me it's not the criterion of classification.

What is this criterion then?

Tirrel