# What happens when a symmetry element is added or removed from a point group? [closed]

What new point group is formed if symmetry element i is added to C3 point group?

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• Then it's not a group. – Zhe Feb 3 at 13:30
• @Zhe the book answer says that S6 point group is formed – rishabhx64 Feb 3 at 15:39
• A group plus one extra element is usually not a group at all. A group plus one extra element and the elements produced by it is usually another, bigger group (in this case $S_6$). – Ivan Neretin Feb 3 at 17:09
• If it isn't obvious right away, we may have to write down all elements of the new group, one by one. – Ivan Neretin Feb 3 at 18:32
• Forget the points, work with matrices. – Ivan Neretin Feb 3 at 19:13

Inversion $$i$$ is equivalent to a two-fold improper rotation $$S_2$$. Introducing a $$2n$$-fold improper rotation axis to the existing $$C_n$$ point group will result in a new $$S_{2n}$$ point group, e.g. in your case $$C_3 ⊗ i = S_6$$.
• @andselisk take this case add $i$ to $C_{3v}$ – rishabhx64 Feb 3 at 18:26