Note: there is another very different and very good introduction to group theory here. Check it out!
Symmetry operations are acts of rotating or reflecting a molecule or figure. Symmetry elements are the symmetry operations that can be done on a molecule to leave it indistinguishable from the way it was when you started.
If we say that a molecule has symmetry C4 about a given axis, we mean that 4 equal rotations (of 90 degrees each) about that axis will get us back to where we started, and that each of the four rotations will leave the molecule in a position identical to its starting position.
Cyclobutane has a C4 axis coming out of the page. If Java is enabled on your browser, you can rotate the figure below by clicking and dragging with your mouse. Notice also the reflection plane (reflection planes are denoted with the Greek letter s, or sigma) that starts out in the plane of the molecule. Move the figure around see if you can make the atoms line up so that the reflection plane is perpendicular to the page.
Note that in these interactive figures, the bonds are not shown. Only the atoms are there. I have colored carbon red and hydrogen blue for your viewing pleasure.
You should learn to recognize the different ways that chemists represent three-dimensional structures on two-dimensional paper. Here is a drawing of cyclobutane as a chemist might sketch it. Notice that the bold lines indicate bonds that come out of the page and the dashed lined indicate bonds that go into the page. This shows that the four carbon atoms approximately define a plane; the hydrogens are on either side of the plane. Even though carbons are singly-bonded to each other, there is not free rotation about this carbon-carbon bond because there isn't room for the hydrogens to squeeze past each other though the ring.
If a molecule has symmetry C3 about some axis, we mean that three equal rotations (of 120 degrees each) about that axis will do the same thing. Methane, a tetrahedral molecule, is shown below. Can you find its C3 axes? Use your mouse to turn the molecule so that one of the blue hydrogens is pointing toward you. Now do you see the C3 axis? Methane also has 6 reflection planes. Each reflection plane is defined by the central carbon and two of the hydrogens.
Here are some alternate representations of methane. The first shows explicitly that the four lobes of the tetrahedron point toward the corners of a cube. The other two use the conventions that bold means out of the page and dashed means into the page. The drawing to the far right has one of the C3 axes of methane in the plane of the page.
All symmetry operations are some combination of rotations and reflections. However, sometimes we do what is called an "improper rotation" in which we rotate the molecule and then reflect it through a plane perpendicular to the axis of the rotation. So, for example, an S4 is just like a C4 except that after rotating the molecule 90 degrees about some axis, we reflect it through a plane perpendicular to that axis. The methane molecule above has an S4. The Official Chem 32 Theme Molecule, allene, also has an S4. Allene is pictured below. Notice that if you were to replace just one of the hydrogens with a heteroatom such as chlorine, the S4 symmetry is destroyed.
One of the reasons allene is so popular in this class is that it is not only interesting from a group theoretical perspective, but also from a bonding perspective. The central carbon is sp hybridized. Its two unhybridized p orbitals are required by quantum mechanical laws to be perpendicular to the sp hybrid orbitals and to each other. The consequence is that the sp hybrid orbitals on the central carbon form sigma bonds with the two outer carbons, one pi bond is formed in the plane of the page, and the other pi bond is perpendicular to the plane of the page. Below is a projection down the carbon-carbon-carbon bonds of allene and a picture of the p orbitals that are involved in its bonding scheme. Notice that you can only see the tops of the p orbitals on the left; they are coming out of the page. This type of system, in which two adjacent pi bonds are formed from perpendicular sets of orbitals, is called a cumulated system and is somewhat unstable (although allenes are found in nature). Later we will contrast such systems with their more stable, conjugated counterparts.
Chemists often talk about "inverting" a molecule (or orbital) or of its having even or odd "inversion symmetry." A molecule that is even to inversion has the property that if you take every point in the molecule, pass it through the molecule's center, and bring it out the same distance on the other side, the molecule will be unchanged. An orbital that is odd to inversion has the property that if you take every point of that orbital, pass it through the orbital's center, and bring it out the same distance on the other side, the sign of the orbital's wavefunction will be inverted (i.e. its shading will be flipped). "Inverting" a molecule is the same as doing an S2 improper rotation. Notice that a simple reflection is also the same as doing an S1 improper rotation. These two facts become especially useful when we talk about chirality. Chiral molecules are defined as molecules that have no improper rotation axes. So if a molecule has reflection or inversion symmetry in any of its conformations, then you immediately can tell that the molecule is not chiral. All of the moveable molecules featured above do have reflection or inversion symmetry, so none of them are chiral.
You may have heard of chiral molecules before and been told that a chiral molecule is a molecule having a carbon with four different substituents (such a carbon is often called a "chiral center"). This is usually a good rule of thumb, good enough for most non-chem 32 students anyway!! However, now that you know the group theoretical definition of chirality, you know that there is more to it than that. For example, look at the molecule of 1,2-dibromo 1,2-dichloroethane below. Each of the carbon atoms has four different substituents. However, since the carbon-carbon bond is a single bond, there is free rotation about that bond. You could line up the atoms to show a mirror plane or an inversion center as in the pictures below. If a molecule has reflection or inversion symmetry in any of its conformations, it is achiral. When we look down the carbon-carbon bond as in the bottom figures (these are called "Newman projections") the inversion symmetry in the "staggered conformer" and the mirror reflection plane in the "eclipsed conformer" are especially evident.
And while we're dispelling introductory chemistry myths, what about the substituted allene molecule below? This is a molecule that most certainly does not possess a carbon with four different substituents--each carbon has a double bond!! However, it does not possess reflection symmetry, inversion symmetry, or any other higher-order improper rotation axes either. This is a chiral molecule without a chiral center.
(It is worth noting on the side that just because you don't see any reflection planes or inversion centers, you cannot immediately say a molecule is chiral. There might be a higher-order improper rotation axis that renders the molecule achiral. This is extremely rare.)