Graf Konjugasi dari Hasil Kali Langsung Grup Alternating A4 dan Grup Simetri S3

On the number of conjugacy classes in a finite group

In this work we obtain new properties connected with the number of conjugacy classes of elements of a finite group, through the analysis of the numberr G(gN) of conjugacy classes of elements ofG that intersect the cosetgN, whereN is a normal subgroup ofG andg any element ofG. The results obtained about this number are not only used in the general problem of classifying finite groups according to the number of conjugacy classes, but they also allow us to improve and generalize known results relating to conjugacy classes due to P. Hall, M. Cartwright, A. Mann, G. Sherman, A. Vera-Lopez and L. Ortiz de Elguea. Examples are given which illustrate our improvements.

Conjugacy Classes in Maximal Parabolic Subgroups of General Linear Groups

In noncommutative geometry a ‘Lie algebra’ or bidirectional bicovariant differential calculus on a finite group is provided by a choice of an ad-stable generating subset$\mathcal{C}$stable under inversion. We study the associated Killing formK. For the universal calculus associated to$\mathcal{C}$=G\ {e} we show that the magnitude$\mu=\sum_{a,b\in\mathcal{C}}(K^{-1})_{a,b}$of the Killing form is defined for all finite groups (even whenKis not invertible) and that a finite group is Roth, meaning its conjugation representation contains every irreducible,iffμ ≠ 1/(N− 1) whereNis the number of conjugacy classes. We show further that the Killing form is invertible in the Roth case, and that the Killing form restricted to the (N− 1)-dimensional subspace of invariant vectors is invertibleiffthe finite group is an almost-Roth group (meaning its conjugation representation has at most one missing irreducible). It is known [9, 10] that most nonabelian finite simple groups are Roth and that all are almost Roth. At the other extreme from the universal calculus we prove that the 2-cycles conjugacy class in anySnhas invertible Killing form, and the same for the generating conjugacy classes in the case of the dihedral groupsD2nwithnodd. We verify invertibility of the Killing forms of all real conjugacy classes in all nonabelian finite simple groups to order 75,000, by computer, and we conjecture this to extend to all nonabelian finite simple groups.

A famous open problem due to Graham Higman asks if the number of conjugacy classes in the group of n × n unipotent upper triangular matrices over the q-element field can be expressed as a polynomial function of q for every fixed n. We consider the generalization of the problem for pattern groups and prove that for some pattern groups of nilpotency class two the number of conjugacy classes is not a polynomial function of q.

Let κ be a characteristic p finite field of q elements and 𝔑 κ {\mathfrak{N}_{\kappa}} the Nottingham group over κ. Lubin associated to every conjugacy class of torsion element of 𝔑 κ {\mathfrak{N}_{\kappa}} a type. We establish an upper bound B ⁢ ( q ; l , m ) {B(q;l,m)} on the number of conjugacy classes of order p 2 {p^{2}} torsion elements u of 𝔑 κ {\mathfrak{N}_{\kappa}} of type 〈 l , m 〉 {\langle l,m\rangle} . In the case where l < p {l<p} , the bound B ⁢ ( q ; l , m ) {B(q;l,m)} is the exact number of conjugacy classes. Moreover, we give a criterion on when u and u n {u^{n}} are conjugate.

SynopsisIn this paper, the number of conjugacy classes in a finite group G is analysed in terms of the number of ordered pairs that generate it. Using this relation, we give a new elementary proof of one of A. Mann's results for finite groups, namely: |G| ≡ r(G) (mod. d|G|. δ|G|), where , prime and pi ≠ Pj for every i≠j, r(G) denotes the number of conjugacy classes of elements of G, d|G| = g.c.d. (p1 − 1, … pt − 1) and δ|G| = g.c.d. . The above congruence is obtained without using character theory. We also obtain new local congruences that slightly improve Mann's congruence.

In this paper, number of conjugacy classes and irreducible characters in a non-abelian group of order $2^6$ are investigated using cycle pattern of elements. Through the exploits of commutator and representation of elements as a product of disjoint cycles, the number of conjugacy classes is obtained which extends some results in literature.

Let V be a complex representation of a finite group G of order g. Derksen conjectured that the pth syzygies of the invariant ring Sym(V ) are generated in degrees ≤ (p+ 1)g. We point out that a simple application of the theory of twisted commutative algebras — using an idea due to Weyl — gives the weaker bound pg, almost for free. Fix a finite group G of order g. Let V be a finite dimensional complex representation of G and put R = Sym(V )G, which we regard as a graded ring. Let E ⊂ R be a homogeneous vector subspace generating R as an algebra, and put S = Sym(E), so that S → R is a surjection of graded algebras. The space Torp (R,C) is then a graded vector space, called the space of p-syzygies of R. We let sp(V ;E) be the maximal degree occurring in it. One can show that if E ⊂ E′ then sp(V ;E) ≤ sp(V ;E′). Furthermore, sp(V ;E) is independent of E if E is chosen to be minimal; we denote this common value by sp(V ). We then have the following conjecture of Derksen (see [D, Conj. 3] for a more precise version): Conjecture 1. We have sp(V ) ≤ (p+ 1)g for any V . Derksen [D, Thm. 2] proved the conjecture for p = 1, but the general case is open. To state our main result, we first introduce some notation. Let β(V ) be the minimal integer such that R is generated in degrees ≤ β(V ). Let β be the maximum value of β(V ) over all V , the so-called Noether number of G. Noether’s theorem [W, §3] states that β ≤ g. Let d1, . . . , dn be the degrees of the irreducible representations of G and let m be the sum of the di. Finally, put δp = (β − 1)g − (m− 1)βp; note that this is negative for p 0. We then have: Theorem 2. We have sp(V ) ≤ β2mp+ δp for any V . Although this is weaker than the conjecture, the bound is significant since it is independent of V . Using the fact that β and m are both bounded by g, we deduce the following corollary: Corollary 3. We have sp(V ) ≤ pg3 for any V . Remark 4. In fact, m and β are often strictly smaller than g, sometimes significantly so. We have m ≤ √ng, where n is the number of conjugacy classes in G. Typically, n is much smaller than g; for example, if G is a symmetric group then n = O(g ), for any > 0. The currently known bounds on β are not very sharp. A result of Cziszter–Domokos [CD] states that β < 1 2g unless G has a cyclic subgroup of index two, or is one of four exceptions. A conjecture of Pawale [W, Conj. 3.9] asserts that if G is the semi-direct product of cyclic groups of prime orders p and q, with q | p− 1, then β = p+ q − 1; note that m = p+ q − 1 in this case as well. When p and q are approximately equal, this gives m = β = O( √ g). Thus, in many case, our theorem gives a bound of the form sp(V ) = O(pg θ) with θ < 3. Remark 5. Theorem 2 (and our proof of it) is valid over any field of characteristic zero. Derksen also proposed his conjecture in positive characteristic not dividing g. It would be interesting if our proof could be adapted to work in this setting. We now prove the theorem. Put sp(V ) = sp(V ;E) where E = ⊕β i=1Ri. As discussed, we have sp(V ) ≤ sp(V ), so it suffices to bound the latter. The key result is the following lemma. Date: November 2, 2012. The author was supported by NSF fellowship DMS-0902661. 1

The aim of this paper is to show how the number of conjugacy classes appearing in the product of classes affect the structure of a finite group. The aim of this paper was to show several results about solvability concerning the case in which the power of a conjugacy class is a union of one or two conjugacy classes. Moreover, we show that the above conditions can be determined through the character table of the group.

The main result of this article is that if a 3-manifold M supports an Anosov flow, then the number of conjugacy classes in the fundamental group of M grows exponentially fast with the length of the shortest orbit representative, hereby answering a question raised by Plante and Thurston in 1972. In fact we show that, when the flow is transitive, the exponential growth rate is exactly the topological entropy of the flow. We also show that taking only the shortest orbit representatives in each conjugacy classes still yields Bowen’s version of the measure of maximal entropy. These results are achieved by obtaining counting results on the growth rate of the number of periodic orbits inside a free homotopy class . In the first part of the article, we also construct many examples of Anosov flows having some finite and some infinite free homotopy classes of periodic orbits, and we also give a characterization of algebraic Anosov flows as the only \mathbb R -covered Anosov flows up to orbit equivalence and finite lifts that do not admit at least one infinite free homotopy class of periodic orbits.

Up to isomorphism, there are only finitely many finite groups with a given number of conjugacy classes. Those with up to twelve classes have already been classified. In this work we extend the classification to thirteen and fourteen classes. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

In this paper we give new information about the conjugacy vector of the group , the Sylow p-subgroup of GL(n, q) consisting of the upper unitriangular matrices. The first two components of this vector are given in [4]. Here, we obtain the third component, that is, the number of conjugacy classes whose centralizer has qn+l elements. Besides, we give the whole set of numbers which compose this vector:

The generalized Pauli group and its normalizer, the Clifford group, have a rich mathematical structure which is relevant to the problem of constructing symmetric informationally complete POVMs (SIC-POVMs). To date, almost every known SIC-POVM fiducial vector is an eigenstate of a "canonical" unitary in the Clifford group. I show that every canonical unitary in prime dimensions p > 3 lies in the same conjugacy class of the Clifford group and give a class representative for all such dimensions. It follows that if even one such SIC-POVM fiducial vector is an eigenvector of such a unitary, then all of them are (for a given such dimension). I also conjecture that in all dimensions d, the number of conjugacy classes is bounded above by 3 and depends only on d mod 9, and I support this claim with computer computations in all dimensions < 48.

We show that the enumeration of linear orbits and conjugacy classes of Z \mathbf {Z} -defined unipotent groups over finite fields is “wild” in the following sense: given an arbitrary scheme Y Y of finite type over Z \mathbf {Z} and integer n ⩾ 1 n\!\geqslant \! 1 , the numbers # Y ( F q ) mod q n \#Y(\mathbf {F}_q) \bmod q^n can be expressed, uniformly in q q , in terms of the numbers of linear orbits (or numbers of conjugacy classes) of finitely many Z \mathbf {Z} -defined unipotent groups over F q \mathbf {F}_q and finitely many Laurent polynomials in 𝑞.