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Consider the group of symmetries of a line and let $I$ be the trivial symmetry

  1. Is this group commutative?
  2. Is there a symmetry $a\neq I$ such that $a^3 = I$?
  3. If you combine a translation and a rotation do you get new kind of symmetry or another kind of rotation or translation?
  4. Is this group isomorphic to the group of symmetries of a circle?

This is very confusing, I know that I can use the number line as an example but how do I do that?

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  • $\begingroup$ What is a rotation if we don't have a plane, but only a line? $\endgroup$ – Ivan Neretin Nov 8 at 6:10
  • $\begingroup$ if we pick a point on the number line then there will be only one rotation which is 180 degree . 360 degree will go back to itself with is the identity $\endgroup$ – Yasameen Nov 8 at 6:19
  • $\begingroup$ I'd rather call that a reflection, though technically you are right. $\endgroup$ – Ivan Neretin Nov 8 at 6:21

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