Group automorphism

On the other hand, every natural number is the order of the automorphism group of a graph with connectivity number 1. 

Sp = 1. Thus N(C) is the sum of the coefficients in the expression C. For a cycle index A we have, by definition, N(A) = 1, From what has been stated above, it follows that N(A B) will be the num- ber of superpositions of the two graphs. Moreover the "product" A B can be extended, by associativity, to any number of cycle indexes, and will again be the cycle index sum for the superpositions of many graphs. Hence if A- denotes, for brevity, the cycle index of the automorphism group of Tj -- our previous Z(C.) --then the number of superposed graphs is given by... 

The distinct superpositions are shown in Figure 6 together with their automorphism groups, from which we can verify the assertions made above about the sum of their cycle indexes. For... 

For our present purpose we shall need to retain much more information about these graphs. Specifically, we want to find the sum of the cycle indexes of their automorphism groups. This is still basically a Polya-type problem, for which we replace T(x) by the sum of the cycle indexes of rooted trees. If T denotes the set of rooted trees, then this cycle index sum can be written Z(T ). Note that we can always recover F(x) from Z( T) for since the sum of the coefficients in the cycle index is 1, we have only to replace each occurrence of 5j by x Each cycle index for a tree on n vertices then reduces to x". This result is general and applies to any cycle index sum. 

SheJ68 Sheehan, J. The number of graphs with a given automorphism group. Canad. J. Math. 20 (1968) 1068-1076. 

StoP71 Stockmeyer, P. K. Enumeration of graphs with prescribed automorphism group. Ph.D. Thesis, U. of Michigan 1971. 

WhiD75 White, D. E. Classifying patterns by automorphism group an operator theoretic approach. Discrete Math. 13 (1975) 277-295. 

WhiD75c White, D. E. counting patterns with a given automorphism group. Proc. Amer. Math. Soc. 47 (1975) 41-44. 

Classification of inequivalent subalgebras of the algebras p(l,3), p(1.3), c(l,3) within actions of different automorphism groups [including the groups P(l, 3), P(l, 3) and 0(1,3)] is already available [30]. Since we will concentrate on conformally invariant systems, it is natural to restrict our disscussion to the classification of subalgebras of c(l, 3) that are inequivalent within the action of the conformal group 0(1, 3). 

Assertion 1. The list of subalgebras of the algebra p, 3) of the rank 3, defined within the action of the inner automorphism group of the algebra c(l,3), is exhausted by the following subalgebras ... 

Only r-gons and non-Platonic plane tilings r, q are isotoxal their respective automorphism groups are Crv and T 2, r, q). The group Aut([r, q] — f) is Crv in five Platonic cases none is isotoxal, isogonal, or isohedral polycycle, except of isohedral 3,3 — / = (3,3)-star. 

Proofs Consider an isohedral (r, )-polycycle P and fix a face F in it The automorphism group of an r-gon is the dihedral group Cr the stabilizer Stab(F) of F in P is a subgroup of Crv. Since Crv is finite, we have a finite number of possibilities... 

Proof. Hike such a polycycle. Its universal cover is an elementary (5,3)-polycycle, whose automorphism group contains some fixed-point-free transformation. Inspection of the list of elementary (5,3)-polycycles in Figure 7.2 yields only Eg = snub Prismoo as a possibility. Snub Prismm is obtained from the group of translations by m faces and the non-orientable quotients if the group contains also some translation followed by reflection. ... 

Let us now determine all elementary ( 2,3,4,5), 3)gOT-polycycles. Hie universal cover P of such a polycycle P is an elementary ( 2,3,4,5, 3)-polycycle, whichhas a non-trivial fixed-point-free automorphism group in Aut(P). Consideration of the above list of polycycles yields snub Prisma, as the only possibility. The polycycles snub Prismm and its non-orientable quotients arise in this process. ... 

If P is an elementary ( 2,3], 5)ge -polycycle, which is not a ( 2,3, 5)-polycycle, then its universal cover P is an elementary ( 2,3, 5)-polycycle, which has a fixed-point-free automorphism group included in Aut(P). Clearly, only snub APrismoo is such and it yields the infinite series of snub APrisnim and its non-orientable quotients. ... 

We obtain a ( 5,8 + , 3)-plane that is (8 + ri)R2. All above planes are periodic hence, by taking the quotient (by a translation subgroup of their automorphism group), we obtain ( 5, b, 3)-tori that are bR2. 

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