Extraspecial Group Research Articles

Categorical Tori

We give explicit and elementary constructions of the categorical extensions of a torus by the circle and discuss an application to loop group extensions. Examples include maximal tori of simple and simply connected compact Lie groups and the tori associated to the Leech and Niemeyer lattices. We obtain the extraspecial 2-groups as the isomorphism classes of categorical fixed points under an involution action.

A structure theorem for product sets in extra special groups

Hegyvári and Hennecart showed that if B is a sufficiently large brick of a Heisenberg group, then the product set B⋅B contains many cosets of the center of the group. We give a new, robust proof of this theorem that extends to all extra special groups as well as to a large family of quasigroups.

Permutability of injectors with a central socle in a finite solvable group

In response to an Open Question of Doerk and Hawkes [5, IX Section 3, page 615], we shall show that if Zπ is the Fitting class formed by the finite solvable groups whose π-socle is central (where π is a set of prime numbers), then the Zπ-injectors of a finite solvable group G permute with the members of a Sylow basis in G. The proof depends on the properties of certain extraspecial groups [4].

Novel Noncommutative Cryptography Scheme Using Extra Special Group

Noncommutative cryptography (NCC) is truly a fascinating area with great hope of advancing performance and security for high end applications. It provides a high level of safety measures. The basis of this group is established on the hidden subgroup or subfield problem (HSP). The major focus in this manuscript is to establish the cryptographic schemes on the extra special group (ESG). ESG is showing one of the most appropriate noncommutative platforms for the solution of an open problem. The working principle is based on the random polynomials chosen by the communicating parties to secure key exchange, encryption-decryption, and authentication schemes. This group supports Heisenberg, dihedral order, and quaternion group. Further, this is enhanced from the general group elements to equivalent ring elements, known by the monomials generations for the cryptographic schemes. In this regard, special or peculiar matrices show the potential advantages. The projected approach is exclusively based on the typical sparse matrices, and an analysis report is presented fulfilling the central cryptographic requirements. The order of this group is more challenging to assail like length based, automorphism, and brute-force attacks.

Regular Orbits of Extra-special Groups

In this paper, for odd primes p we find a recursive formula for the number of regular orbits of an extra-special p-group of exponent p2 acting on a faithful irreducible module over a finite field. This complements an earlier result of Foulser for extra-special groups of order p.

Existence of non-abelian representations of the near hexagon Q(5,2)⊗Q(5,2)

In [5], a new combinatorial model with four types of points and nine types of lines of the slim dense near hexagon Q(5,2)⊗Q(5,2) was provided and it was then used to construct a non-abelain representation of Q(5,2)⊗Q(5,2) in the extraspecial 2-group \(2_{-}^{1+18}\). In this paper, we give a direct proof for the existence of a non-abelian representation of Q(5,2)⊗Q(5,2) in \(2_{-}^{1+18}\).

Commutative Schur Rings of Maximal Dimension

A commutative Schur ring over a finite group G has dimension at most s G = d 1 + … +d r , where the d i are the degrees of the irreducible characters of G. We find families of groups that have S-rings that realize this bound, including the groups SL(2, 2 n ), metacyclic groups, extraspecial groups, and groups all of whose character degrees are 1 or a fixed prime. We also give families of groups that do not realize this bound. We show that the class of groups that have S-rings that realize this bound is invariant under taking quotients. We also show how such S-rings determine a random walk on the group and how the generating function for such a random walk can be calculated using the group determinant.

Pentavalent Symmetric Graphs of Order Twice a Prime Square

A classification of pentavalent symmetric graphs of order twice a prime square is given. It is proved that such a graph is a coset graph of ℤ3. A 6 (non-split extension), or a bi-coset graph of an extra-special group of order 125, or the standard double cover of a specific abelian Cayley digraph of order a prime square.

On fixed-point-free automorphisms

Let R be a cyclic group of prime order which acts on the extraspecial group F in such a way that F=[F,R]. Suppose RF acts on a group G so that CG(F)=1 and (|R|,|G|)=1. It is proved that F(CG(R))⊆F(G). As corollaries to this, it is shown that the Fitting series of CG(R) coincides with the intersections of CG(R) with the Fitting series of G, and that when |R| is not a Fermat prime, the Fitting heights of CG(R) and G are equal.

Homotopy colimits of classifying spaces of abelian subgroups of a finite group

The classifying space BG of a topological group $G$ can be filtered by a sequence of subspaces $B(q,G)$, using the descending central series of free groups. If $G$ is finite, describing them as homotopy colimits is convenient when applying homotopy theoretic methods. In this paper we introduce natural subspaces $B(q,G)_p$ of $B(q,G)$ defined for a fixed prime $p$. We show that $B(q,G)$ is stably homotopy equivalent to a wedge of $B(q,G)_p$ as $p$ runs over the primes dividing the order of $G$. Colimits of abelian groups play an important role in understanding the homotopy type of these spaces. Extraspecial $2$-groups are key examples, for which these colimits turn out to be finite. We prove that for extraspecial 2-groups, $B(2,G)$ does not have the homotopy type of a $K(\pi,1)$ space. For a finite group $G$, we compute the complex K-theory of $B(2,G)$ modulo torsion.

On the Existence of Johnson Polynomials for Nilpotent Groups

Let G be a finite group. We say that G has a Johnson polynomial if there exists a polynomial f(x) ∈ ℤ[x] and a character χ ∈ Irr(G) so that f(χ) equals the total character for G. In this paper, we show that if G has nilpotence class 2, then G has a Johnson polynomial if and only if G is an extra-special 2-group. Generalizing this, we say that G has a generalized Johnson polynomial if f(x) ∈ ℚ[x]. We show that if G has nilpotence class 2, then G has a generalized Johnson polynomial if and only if Z(G) is cyclic. Also, if G is nilpotent and |cd(G)| = 2, then G has a generalized Johnson polynomial if and only if G has nilpotence class 2 and Z(G) is cyclic.

A construction of a partial difference set in the extraspecial groups of order $$p^3$$ p 3 with exponent $$p^2$$ p 2

A partial difference set $$S$$ S in a finite group $$G$$ G such that $$S = S^{-1}$$ S = S - 1 and $$1 \notin S$$ 1 ? S corresponds to an undirected strongly regular Cayley graph $$\mathrm{Cay}(G,S).$$ Cay ( G , S ) . Very few examples of PDSs are known, and there are especially few known in nonabelian groups. In this paper, a partial difference set $$S$$ S with $$S = S^{-1}$$ S = S - 1 and $$1 \notin S$$ 1 ? S is constructed for each extraspecial group of order $$p^3$$ p 3 and exponent $$p^2$$ p 2 , where $$p$$ p is an odd prime.

Fields of algebraic numbers with bounded local degrees and their properties

We provide a characterization of infinite algebraic Galois extensions of the rationals with uniformly bounded local degrees, giving a detailed proof of all the results announced in a paper by Checcoli and Zannier and obtaining relevant generalizations for them. In particular we show that that for an infinite Galois extension of the rationals the following three properties are equivalent: having uniformly bounded local degrees at every prime; having uniformly bounded local degrees at almost every prime; having Galois group of finite exponent. The proof of this result enlightens interesting connections with Zelmanov's work on the Restricted Burnside Problem. We give a formula to explicitly compute bounds for the local degrees of a infinite extension in some special cases. We relate the uniform boundedness of the local degrees to other properties: being a subfield of the compositum of all number fields of bounded degree; being generated by elements of bounded degree. We prove that the above properties are equivalent for abelian extensions, but not in general; we provide counterexamples based on group-theoretical constructions with extraspecial groups and their modules, for which we give explicit realizations.

On the descending central sequence of absolute Galois groups

Let $p$ be an odd prime number and $F$ a field containing a primitive $p$th root of unity. We prove a new restriction on the group-theoretic structure of the absolute Galois group $G_F$ of $F$. Namely, the third subgroup $G_F^{(3)}$ in the descending $p$-central sequence of $G_F$ is the intersection of all open normal subgroups $N$ such that $G_F/N$ is $1$, ${\Bbb Z}/p^2$, or the extra-special group $M_{p^3}$ of order $p^3$ and exponent $p^2$.

A representation of extra-special 2-group, entanglement, and Berry phase of two qubits in Yang-Baxter system

In this paper we show another representations of extra-special 2-groups. Based on this new representation, we infer a \({\mathbb{M}}\) matrix which obeys the extra-special 2-groups algebra relations. We also derive a unitary \({\breve{R}(\theta,\varphi)}\) matrix from the \({\mathbb{M}}\) using the Yang-Baxterization process. A Hamiltonian for the two qubits is constructed from the unitary \({\breve{R}(\theta,\varphi)}\) matrix. In this way, we study the Berry phase and entanglement of the two-qubit system. The results also establish relations between topological and holonomic quantum computation.

Minimal characteristic bisets and finite groups realizing Ruiz–Viruel exotic fusion systems

Continuing our previous work (Park, 2010) [2], we determine a minimal left characteristic biset X for every exotic fusion system F on the extraspecial group S of order 7 3 and exponent 7 discovered by Ruiz and Viruel (2004) [6], and analyze the finite group G obtained from X by the method of Park (2010) [2] which realizes F as the full subcategory of the 7-fusion system of G. In particular, we obtain an upper bound for the exoticity index for F .

An Efficient Quantum Algorithm for the Hidden Subgroup Problem in Nil-2 Groups

In this paper we show that the hidden subgroup problem in nil-2 groups, that is in groups of nilpotency class at most 2, can be solved efficiently by a quantum procedure. The algorithm is an extension of our earlier method for extraspecial groups in Ivanyos et al. (Proceedings of the 24th Symposium on Theoretical Aspects of Computer Science (STACS), vol. 4393, pp. 586–597, 2007), but it has several additional features. It contains a powerful classical reduction for the hidden subgroup problem in nilpotent groups of constant nilpotency class to the specific case where the group is a p-group of exponent p and the subgroup is either trivial or cyclic. This reduction might also be useful for dealing with groups of higher nilpotency class. The quantum part of the algorithm uses well chosen group actions based on some automorphisms of nil-2 groups. The right choice of the actions requires the solution of a system of quadratic and linear equations. The existence of a solution is guaranteed by the Chevalley-Warning theorem, and we prove that it can also be found efficiently.

Imaginary transformations

We investigate classes of discrete musical operations that function in ways that correspond to the imaginary units i, j, and k of the quaternions (i.e. orthogonal square roots of−1). In particular, we focus on musical transformations of order 4 that are mutual square roots of a central involution. We define imaginary transformations as members of groups of operations acting on the set of pitch classes in an octatonic collection and as members of subgroups of various triadic-transformational systems. These structures include quaternion, Pauli, (almost) extraspecial, and dicyclic groups, among others.

Extraspecial two-Groups, generalized Yang-Baxter equations and braiding quantum gates

In this paper we describe connections among extraspecial 2-groups, unitary representations of the braid group and multi-qubit braiding quantum gates. We first construct new representations of extraspecial 2-groups. Extending the latter by the symmetric group, we construct new unitary braid representations, which are solutions to generalized Yang-Baxter equations and use them to realize new braiding quantum gates. These gates generate the GHZ (Greenberger-Horne-Zeilinger) states, for an arbitrary (particularly an \emph{odd}) number of qubits, from the product basis. We also discuss the Yang-Baxterization of the new braid group representations, which describes unitary evolution of the GHZ states. Our study suggests that through their connection with braiding gates, extraspecial 2-groups and the GHZ states may play an important role in quantum error correction and topological quantum computing.

Extraspecial two-groups, generalized yang-baxter equations and braiding quantum gates

In this paper we describe connections among extraspecial 2-groups, unitary representa-tions of the braid group and multi-qubit braiding quantum gates. We first construct newrepresentations of extraspecial 2-groups. Extending the latter by the symmetric group,we construct new unitary braid representations, which are solutions to generalized Yang-Baxter equations and use them to realize new braiding quantum gates. These gates gen-erate the GHZ (Greenberger-Horne-Zeilinger) states, for an arbitrary (particularly anodd) number of qubits, from the product basis. We also discuss the Yang-Baxterizationof the new braid group representations, which describes unitary evolution of the GHZstates. Our study suggests that through their connection with braiding gates, extraspe-cial 2-groups and the GHZ states may play an important role in quantum error correctionand topological quantum computing.