In this paper, we count the number of fuzzy subgroups of cyclic group G = Zp × Zq × Zr × Zs with p, q, r, s are distinct prime numbers. We define an equivalence relation on the fuzzy subgroups of any groups G and determine all of its subgroups. Then we draw the diagram of lattice subgroups of G. The number of fuzzy subgroups of G by observing the diagram then will be counted. Mathematics Subjec...

Let Mod(S) be the extended mapping class group of a surface S. For S the twice-punctured torus, we show that there exists an isomorphism of finite index subgroups of Mod(S) which is not the restriction of any inner automorphism. For S a torus with at least three punctures, we show that every injection of a finite index subgroup of Mod(S) into Mod(S) is the restriction of an inner automorphism o...

Although Pilch-Warner (PW) gravitational renormalization group flow [1] passes a number of important consistency checks to be identified as a holographic dual to a largeN SU(N) N = 2 supersymmetric gauge theory, it fails to reproduce the free energy of the theory on S, computed with the localization techniques. This disagreement points to the existence of a larger dual gravitational consistent ...

We provide an application of the theory group actions to study musical scales. For any G, finite G-set S, and real number t, we define t-power diameter diamt(G,S) be size maximal orbit S divided by mean elements S. The symmetric S11 acts on set all tonic scales, where a scale is subset Z12 containing 0. show that for t∈[−1,1], among subgroups G S11, heptatonic scales largest subgroup Γ, its con...

Let Homeo(S) represent the full group of homeomorphisms of the unit circle S, and let A represent the set of subgroups of Homeo(S) satisfying the two properties that if G ∈ A then 1) G contains only orientationpreserving homeomorphisms of S and 2) G contains no non-abelian free subgroups. In this article we use classical results about homeomorphisms of the circle and elementary dynamical method...