Main [14]. To get a distinctive viewpoint, the readers may perhaps consult Reference [15]. two. Graph Coverings and Conjugacy Classes of a Finitely Generated Group Let rel( x1 , x2 , . . . , xr ) be the relation defining the finitely presented group f p = x1 , x2 , . . . , xr |rel( x1 , x2 , . . . , xr ) on r letters (or generators). We are serious about the conjugacy classes (cc) of subgroups of f p with respect towards the nature of the relation rel. In a nutshell, 1 observes that the cardinality structure d ( f p) of conjugacy classes of subgroups of index d of f p is all of the closer to that on the no cost group Fr-1 on r – 1 generators as the option of rel consists of far more non local structure. To arrive at this statement, we experiment on protein foldings, musical types and poems. The former case was very first explored in [3]. Let X and X be two graphs. A graph epimorphism (an onto or surjective homomor phism) : X X is known as a ML-SA1 site covering projection if, for every vertex v of X, maps the neighborhood of v bijectively onto the neighborhood of v. The graph X is referred to as a base graph (or possibly a quotient graph) and X is named the covering graph. The conjugacy classes of subgroups of index d within the fundamental group of a base graph X are in one-to-one correspondence with the connected d-fold coverings of X, as it has been known for some time [16,17]. Graph coverings and group actions are closely related. Let us get started from an enumeration of integer partitions of d that satisfy:Sci 2021, 3,3 ofl1 2l2 . . . dld = d, a well-known trouble in analytic quantity theory [18,19]. The number of such partitions is p(d) = [1, 2, 3, five, 7, 11, 15, 22 ] when d = [1, 2, 3, four, five, 6, 7, eight ]. The number of d-fold coverings of a graph X from the first Betti number r is ([17], p. 41), Iso( X; d) =l1 2l2 …dld =d(l1 !2l2 l2 ! . . . dld ld !)r-1 .Yet another interpretation of Iso( X; d) is found in ([20], Euqation (12)). Taking a set of mixed quantum states Icosabutate In Vitro comprising r 1 subsystems, Iso( X; d) corresponds to the steady dimension of degree d regional unitary invariants. For two subsystems, r = 1 and such a stable dimension is Iso( X; d) = p(d). A table for Iso( X, d) with modest d’s is in ([17], Table three.1, p. 82) or ([20], Table 1). Then, 1 desires a theorem derived by Hall in 1949 [21] concerning the quantity Nd,r of subgroups of index d in Fr Nd,r = d(d!)r-1 -d -1 i =[(d – i)!]r-1 Ni,rto establish that the number Isoc( X; d) of connected d-fold coverings of a graph X (alias the number of conjugacy classes of subgroups in the fundamental group of X) is as follows ([17], Theorem 3.2, p. 84): Isoc( X; d) = 1 dm|dNm,r d l| md l (r -1) m 1 , mlwhere denotes the number-theoretic M ius function. Table 1 offers the values of Isoc( X; d) for smaller values of r and d ([17], Table 3.two).Table 1. The quantity Isoc( X; d) for smaller values of initially Betti number r (alias the number of generators on the absolutely free group Fr ) and index d. As a result, the columns correspond to the quantity of conjugacy classes of subgroups of index d within the absolutely free group of rank r. r 1 2 three four 5 d=1 1 1 1 1 1 d=2 1 three 7 15 31 d=3 1 7 41 235 1361 d=4 1 26 604 14,120 334,576 d=5 1 97 13,753 1,712,845 207,009,649 d=6 1 624 504,243 371,515,454 268,530,771,271 d=7 1 4163 24,824,785 127,635,996,839 644,969,015,852,The finitely presented groups G = f p may possibly be characterized when it comes to a very first Betti number r. To get a group G, r will be the rank (the number of generators) from the abelian quotient G/[ G, G ]. To some extent, a group f p whose 1st Betti numb.