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In chapter 8.3.5 of the Int. Tables for Cryst. Vol A it is stated that “in group theory, a set of generators of a group is a set of group elements such that each group element may be obtained as an ordered product of the generators”.

According to this definition, the identity should not be required because generators form a set, not a group.

In fact, each of the symmetry operations selected as generator (if we exclude identity) can generate the identity itself; if our generator is g, there will be always a n integer value for which gn = identity (if g is a symmetry operations). For example, if a 2-fold axis along z is a generator (g), identity is generated by applying this symmetry operation twice (n=2); if it is a 4-fold axis, identity is generated by applying this symmetry operation 4 times.

So what is the reason why identity is always included among generators?

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  • $\begingroup$ There is no mathematical reason for that, but it doesn't hurt either. $\endgroup$ Jul 13 at 10:17

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