Enumeration by symmetry

Besides counting orbits by weight there is another refinement, the enumeration by symmetry. For certain applications it is important to evaluate the number of unlabeled asymmetric structures. Our approach uses a theorem from Burnside, which was mentioned briefly above. It is much stronger than the Lemma of Cauchy-Frobenius. 

Let gX denote a finite action. Elements in the seune orbit have conjugate stabilizers since = gGg K Moreover, the set of stabilizers of the elements in the orbit G(x) is the fiiU class G of subgroups conjugate to the stabilizer of x  

Hence the conjugacy class of these stabilizers, in short the stabilizer class, is a characteristic of the orbit. But it should be clear that several orbits can have the same stabilizer class and that different x e X can have the same stabilizer. For example, several permutational isomers may be asymmetric, i.e. G = 1, which means that the conjugacy class of the stabilizer consists of the identity subgroup only, G = 1. Therefore, we introduce the set of orbits with the conjugacy class L7 of a subgroup 17 of G as set of stabilizers for this particular set of orbits, using the following notation  

In fact, the stabilizer class characterizes an orbit up to similarity Let X and qZ denote finite transitive actions, i.e. both G X = 1 and G Z = 1. The following is easy to check  

the orbits with the same stabilizer class are essentially the same , at least as lar as the action is concerned. In this sense, the stabilizer class characterizes the symmetry type of an orbit, 

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