What I've understood about chirality and elements of symmetry:

A molecule that is not superposable on its mirror image is said to be chiral.

Plane of symmetry: An imaginary plane cutting a compound into two symmetric halves. Any compound possessing it is achiral, so any compound without it is chiral?

Axis of symmetry: I have never spent so long trying to understand a concept. An axis of symmetry is an imaginary axis around which by rotating through a minimum angle of rotation, the original compound is obtained.

1. Any compound, be it chiral, achiral or dissymmetric, can have an AOS.

2. You cannot say a compound is achiral just because it has an AOS.

3. Unless, I just had a thought. If on rotation it gives the same compound, making it superposable with the original, thereby making it achiral.

However, wouldn't all compounds (chiral or achiral) on rotation through a particular degree give back the same compound? Please tell me if everything I've mentioned is right.

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    $\begingroup$ Everything you've mentioned is right, except item 3, which contradicts the rest. Indeed, any molecule with axis of symmetry, chiral or not, would coincide with itself if turned for the appropriate angle around that axis. This has nothing to do with chirality. $\endgroup$ – Ivan Neretin May 27 '16 at 15:13

Chirality is left hand—right hand. The left hand is a mirror image of your right hand. If you take this mirror image and then rotate or shift it and you are able to superimpose it onto the original image, it is achiral. This is not possible with the hands, the thumb will always be wrong.

A plane of symmetry is sufficient but not mandatory to give an achiral compound. But it should be clear that mirroring an image that contains an intrinsic mirroring plane will give the same.

Another sufficient but not mandatory symmetry operation to generate a chiral compound is inversion. If some compound is point symmetric, it is never chiral.

And there are other elements of symmetry that are rare if not accompanied by planes or centres of symmetry but which are sufficient for generating achiral compounds: Improper rotation. In a nutshell, this is a symmetry element that requires a rotation with a subsequent mirroring perpendicular to the rotation axis. Take a tetrahedral compound such as methane: If you select the correct axis, rotate by $90^\circ$ and then mirror it perpendicular to that axis, you will receive the same product.

Strictly speaking, planes and centres of symmetry are special cases of improper rotation with a rotation of $360^\circ$ and $180^\circ$, respectively. So one could say that improper rotation is sufficient and mandatory to give an achiral compound.

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  1. is trivially correct since every object has $C_1$ symmetry, i.e. after rotation around an axis about 360 degrees it is indistinguishable from the starting state.

  2. follows from 1.

  3. is not correct. By definition a molecule is chiral if it does not have an improper axis of rotation.

Yes. In general an object stays the same if you look at it from a different point of view.

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First of all it's very ovious that there's no point of comparing the same molecule with itself, because its self depicting! It could exist only for Cn, when n≠1

AOS doesn't assign achirality because Centre of Symmetry (COS) does assign achirality

Reason- An inversion through a center is equivalent to rotating groups by 180° and then reflecting the groups through a plane perpendicular to the rotation axis. If we don't reflect it in the $2nd$ step (i.e. on a plane perpendicular to the Axis of rotation) then it would be the the check for only the AOS. So tell me if COS goes the full way proving that a molecule is achiral then how can AOS do it half the way?

Exceptions are that of highly symmetrical molecules

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