# Abstract Algebra, The group of symmetries of a line.

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?

• What is a rotation if we don't have a plane, but only a line? – Ivan Neretin Nov 8 at 6:10
• 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 – Yasameen Nov 8 at 6:19
• I'd rather call that a reflection, though technically you are right. – Ivan Neretin Nov 8 at 6:21