Finite Nonabelian Group Research Articles

Strongly connective degree sets II: perfect derived subgroups

When G is a finite nonabelian group, we associate the common-divisor graph with G by letting nontrivial degrees in cd(G) = {χ(1) | χ∈Irr(G)} be the vertices and making distinct vertices adjacent if they have a common nontrivial divisor. A set \({\user1{\mathcal{C}}}\) of vertices for this graph is said to be strongly connective for cd(G) if there is some prime which divides every member of \({\user1{\mathcal{C}}}\), and every vertex outside of \({\user1{\mathcal{C}}}\) is adjacent to some member of \({\user1{\mathcal{C}}}\). When G is nonsolvable, we provide sufficiency conditions for cd(G) to have a strongly connective subset. We also extend a previously known result about groups with nonabelian solvable quotients, and prove for arbitrary groups G that if the associated graph is connected and has a diameter bounded by 2, then indeed cd(G) has a strongly connective subset. The major focus is on when the derived subgroup G′ is perfect.

Bent functions on a finite nonabelian group

We introduce the notion of a bent function on a finite nonabelian group which is a natural generalization of the well-known notion of bentness on a finite abelian group due to Logachev, Salnikov and Yashchenko. Using the theory of linear representations and noncommutative harmonic analysis of finite groups we obtain several properties of such functions similar to the corresponding properties of traditional abelian bent functions.

Unfused Involutions in Finite Groups: An Addendum

If p is a prime, r is an integer and p r > 2, there exist infinitely many finite simple non-Abelian groups containing an element x of order p r such that if P is a Sylow p-subgroup of G containing x, x G ∩ P = x P .

Finite simple groups with complemented maximal subgroups

We study Problem 8.31 in [1] of the description of finite weakly factorizable groups, i.e., the groups whose every proper subgroup is complemented in a larger subgroup. By Lemma 1 of [2], the subdirect product of a completely factorizable group (such groups were studied by Ph. Hall and N. V. Baeva (Chernikova)) and a weakly factorizable group is a weakly factorizable group. In connection with a remark in [2], we observe that a dihedral 2-group is always weakly factorizable but, in general, it cannot even be obtained from the groups of prime order by repeated application of the lemma. Theorem 1, basing on the available maximal factorizations, shows that there exist exactly three finite simple nonabelian groups with complemented maximal subgroups. Theorem 2 confirms the conjecture of [2] on uniqueness of a finite simple nonabelian group with the property of weak factorizability. Earlier Theorems 1 and 2 were proven in special cases by A. G. Likharev.

Non-commuting graph of a group

Let G be a non-abelian group and let Z ( G ) be the center of G. Associate a graph Γ G (called non-commuting graph of G) with G as follows: Take G \\ Z ( G ) as the vertices of Γ G and join two distinct vertices x and y, whenever x y ≠ y x . We want to explore how the graph theoretical properties of Γ G can effect on the group theoretical properties of G. We conjecture that if G and H are two non-abelian finite groups such that Γ G ≅ Γ H , then | G | = | H | . Among other results we show that if G is a finite non-abelian nilpotent group and H is a group such that Γ G ≅ Γ H and | G | = | H | , then H is nilpotent.

On the topology of stationary black hole event horizons in higher dimensions

In four dimensions the topology of the event horizon of an asymptotically flat stationary black hole is uniquely determined to be the two-sphere $S^2$. We consider the topology of event horizons in higher dimensions. First, we reconsider Hawking's theorem and show that the integrated Ricci scalar curvature with respect to the induced metric on the event horizon is positive also in higher dimensions. Using this and Thurston's geometric types classification of three-manifolds, we find that the only possible geometric types of event horizons in five dimensions are $S^3$ and $S^2 \times S^1$. In six dimensions we use the requirement that the horizon is cobordant to a four-sphere (topological censorship), Friedman's classification of topological four-manifolds and Donaldson's results on smooth four-manifolds, and show that simply connected event horizons are homeomorphic to $S^4$ or $S^2\times S^2$. We find allowed non-simply connected event horizons $S^3\times S^1$ and $S^2\times \Sigma_g$, and event horizons with finite non-abelian first homotopy group, whose universal cover is $S^4$. Finally, following Smale's results we discuss the classification in dimensions higher than six.

The largest lengths of conjugacy classes and the Sylow subgroups of finite groups

Let G be a finite nonabelian group, P ∈Sylp(G), and bcl(G) the largest length of conjugacy classes of G. In this short paper, we prove that in general and |P/Op(G)| < bcl(G) in the case where P is abelian.

Counting maps from a surface to a graph

Let F be a non-abelian finite rank free group, and let H_g be the fundamental group of a surface of genus g with one boundary component represented by D_g in H_g. So, H_g is the free group and D_g is the product of commutators [a_1,b_1]...[a_g,b_g]. Given x in F, we are interested in the number num(x) of primitive, i.e. root-free, images of monomorphisms (H_g,D_g) -> (F,x). Our main result is that f(g) >= 2^g where f(g)=sup num(x), where sup is taken over all elements x in F. This answers a question of Zlil Sela that is related to his work on the Tarski problem. We also show that f is independent of F and go on to obtain similar results where the monomorphisms considered are additionally required to have minimal genus.

Strongly connective degree sets I: nonabelian solvable quotients

When G is a finite nonabelian group, we associate the common-divisor graph Γ(G) with G by letting nontrivial degrees in cd(G) be the vertices and making distinct vertices adjacent if they have a common nontrivial divisor. A set \(\mathcal{C}\) of vertices for this graph is said to be strongly connective for cd(G) if there is some prime which divides every member of \(\mathcal{C},\) and every vertex outside of \(\mathcal{C}\) is adjacent to some member of \(\mathcal{C}.\) When G has a nonabelian solvable quotient, we show that if Γ(G) is connected and has a diameter of at most 2, then indeed cd(G) has a strongly connective subset.

N-Isoclinism Classes and n-Nilpotency Degree of Finite Groups

Gallagher (1970) and Gustafson (1973) introduced the commutativity degree of a finite group. In this paper, we define the n-nilpotency degree of finite groups for n ≥ 1, and prove some results as Lescot (1995) does for a certain class of groups. In particular, it is shown that the n-isoclinism of finite groups preserves their n-nilpotency degrees. Finally, some sharper and more general upper bound than previously known is constructed for the commutativity degree of non-abelian finite groups.

A simple proof of the Uncertainty Principle for compact groups

We give a simple proof of the Uncertainty Principle for finite nonabelian groups, which generalizes directly to compact groups.

Remarks on systems and differential operators on groups

Complexity and large size of contemporary control, communication, and computer systems impose strong challenges to methods of their mathematical modeling, design, and testing. Classical approaches to their formal specification by exploiting theory of systems on groups of real numbers R and complex numbers C, often do not fulfill requirements of practice. For that reason, theory of systems on groups different from R and C has been developed. Differential operators and spectral (Fourier) analysis on groups play the same important role in such systems as in the case of systems modeled by signals defined on R and C. This paper first briefly reviews some aspects of research in system theory on groups and then presents an extension of the notion of Gibbs differentiation to matrix- valued functions on finite non-Abelian groups.

Classical and quantum function reconstruction via character evaluation

We consider two natural instances of the problem of determining a function f: G→ G defined on a group G by making repeated queries to the oracle O(x)=χ( f(x)) , where χ is a known character of the group G. In particular, we consider the problem of recovering a hidden monic polynomial f( X) of degree d⩾1 over a finite field F p of p elements given a black box which, for any x∈ F p , evaluates the quadratic character of f( x). We design a classical algorithm of complexity O( d 2 p d+ ε ) and also show that the quantum query complexity of this problem is O( d). Some of our results extend those of Wim van Dam, Sean Hallgren and Lawrence Ip obtained in the case of a linear polynomial f( X)= X+ s (with unknown s); some are new even in this case. We then consider the related problem of determining an element s of a finite group G given the oracle O(x)=χ(sx) , where χ is the character of a (known) faithful irreducible representation ρ of G. We show that if d is the dimension of ρ, then O(d 2 log|G|) classical queries suffice to determine s. The proof involves development of a new “uncertainty principle” for the Fourier transform over finite non-Abelian groups.

Wigner distributions for non-Abelian finite groups of odd order

Wigner distributions for quantum mechanical systems whose configuration space is a finite group of odd order are defined so that they correctly reproduce the marginals and have desirable transformation properties under left and right translations. While for the Abelian case we recover known results, though from a different perspective, for the non-Abelian case our results appear to be new.

Finite group theory for large systems. 2. Generating relations and irreducible representations for the icosahedral point group, I(h).

Generating relations and irreducible representations are given for the icosahedral point group, I(h), that are suited to computerized projection of symmetry-adapted bases of arbitrary spaces invariant to the group. With 120 elements I(h ) is a good prototype for symmetry-adaptation to a large finite nonabelian group. The four- and five-dimensional irreducible representations are obtained by coupling direct products of the three-dimensional irreducible representations using the symmetry-adaptation algorithm. The method is applied to the Hückel treatment of icosahedral C(20) fullerene.

Generalized Kramers–Wannier duality for spin systems with non-commutative symmetry

In 1941 H Kramers and G Wannier discovered a special symmetry which relates low-temperature and high-temperature phases in the planar Ising model. The corresponding transformation, the Kramers–Wannier transform, is a special non-local substitution in the partition function. The existence of such transformations is a general property of lattice spin systems. Generalization of the KW transform to spin systems with non-Abelian symmetry is essential for many problems in statistical physics and field theory. This problem is very difficult and cannot be carried out by classical methods (like Fourier transform in the commutative case). We present new results which solve this problem for finite non-Abelian groups.

Finite groups with graphs containing no triangles

Let G be a non-Abelian finite group, Γ( G) the attached graph related to its conjugacy classes. The aim of this paper is to prove that the symmetric group S 3, the dihedral group D 5, the three pairwise non-isomorphic non-Abelian groups of order 12, and the non-Abelian group T 21 of order 21, is the complete list of all G such that Γ( G) contains no triangles.

Polynomial Functions and Endomorphism Near-Rings on Certain Linear Groups

We describe the unary polynomial functions on the non-solvable groups G with SL(n, q) ≤ G ≤ GL(n, q) and on their quotients G/Y with Y ≤ Z(G), and we compute the size of the inner automorphism near-ring I(G/Y). We compare this near-ring to the endomorphism near-ring E(G/Y), and we obtain a full characterization of those G and Y for which I(G/Y) = E(G/Y) holds. For the case Y = {1}, this characterization yields that we have E(G) = I(G) if and only if G = SL(n, q). We investigate the automorphism near-ring A(G), and we show that for all non-solvable groups G with SL(n, q) ≤ G ≤ GL(n, q), we have I(G) = A(G). Our results are based on a description of the polynomial functions on those non-abelian finite groups G that satisfy the following conditions: G′ = G″, G/Z(G) is centerless, and there is no normal subgroup N of G with G′ ∩ Z(G) < N < G′.

Multiplicative Cellular Automata on Nilpotent Groups: Structure, Entropy, and Asymptotics

If \(\mathbb{M} = \mathbb{Z}^D \), and \(\mathcal{B}\) is a finite (nonabelian) group, then \(\mathcal{B}^\mathbb{M}\) is a compact group; a multiplicative cellular automaton (MCA) is a continuous transformation \(\mathfrak{G}:B^\mathbb{M} \to B^\mathbb{M}\) which commutes with all shift maps, and where nearby coordinates are combined using the multiplication operation of \(\mathcal{B}\). We characterize when MCA are group endomorphisms of \(\mathcal{B}^\mathbb{M}\), and show that MCA on \(\mathcal{B}^\mathbb{M}\) inherit a natural structure theory from the structure of \(\mathcal{B}\). We apply this structure theory to compute the measurable entropy of MCA, and to study convergence of initial measures to Haar measure.

More Zariski pairs and finite fundamental groups of curve complements

We give a recipe for finding new examples of plane curves with a finite non-abelian fundamental group of the complement and new examples of Zariski pairs of plane curves.