# Is axis of symmetry considered a true symmetry?

In most of the books when i read about optical activity of a compound this is what they say:

A compound that does not posses any kind of symmetry is optically active. In other words chiral compounds without having any kind of of symmetry are optically active.

However, this holds true when we refer to plane of symmetry or centre of symmetry, because molecules having axis of symmetry or alternate axis of symmetry are still optically active.

For example this compound possesses axis of symmetry but is still optically active: And therefore i am totally confused about relation of optical activity with symmetry.

• It has to be non trivial. A rotation of $0$ or $2\pi$ about an axis being trivial. – K_P Apr 17 '16 at 16:18
• No, this is not what they say. If you skip a few words from a meaningful sentence, it is quite easy to arrive at nonsense. It is not just "any kind of symmetry"; it is "any kind of inversion symmetry", or "any kind of symmetry with determinant -1". Rotational symmetry does not apply, though it is surely true and real (and non-trivial, for that matter). – Ivan Neretin Apr 17 '16 at 17:24

A molecule is chiral (1, 2) if it does not have an improper axis of rotation. Since your drawing does not seem to show the meso form of 2,3-dichlorobutane, it is optically active.

This is to complement the answer given already and to address @Ivan Neretin's comment under OP's question. It is an example that generally (with lots of asterisks) rotations cannot be excluded. This is a copy from J. March's 6ed (p. 158). From the text:

Such compounds possess ... an alternating axis of symmetry as in 1

Of course Ivan is correct that you only care for "determinant $-1$" motions or else orientation reversing and in this case we use a rotation by $\pi$ which has determinant 1 (or else belongs to $SO_3$). My group theory is not as good since I am an organic chemist but I think the actual symmetry element is something like $\rho_{\pi/2} \circ r = 1$ whereas the rotation element above is $\rho_\pi=(\rho_{\pi/2} \circ r)\circ (\rho'_{\pi/2} \circ r')=1 \circ 1=1$ and it does have determinant 1 but is not one of the symmetry group elements (in the sense that $x^{34}$ is in $C_4=\{1,x,x^2,x^3 \}$ but only as $x^{34}=x^2$ ).

To make things more complicated March is defining an "alternating axis of symmetry" as:

An alternating axis of symmetry 17 of order n is an axis such that when an object containing such an axis is rotated by 360/n about the axis and then reflection is effected across a plane at right angles to the axis, a new object is obtained that is indistinguishable from the original one

So it appears that in the example of (1) we talk of $Z_4$ symmetry (if we call $Z$ the alternating axis subgroup) and $Z_4 \sim C_2$

Finally, I think that (1) is an extreme example and almost always for organic compounds chirality (or not) is determined by the lack (or not) of a plane of symmetry

• isn't symmetry of axis a disqualifying property for a molecule being an isomer? – bonCodigo Jul 17 '17 at 3:12