Centralizer Of Involution Research Articles

NOWHERE-ZERO -FLOWS IN TWO FAMILIES OF VERTEX-TRANSITIVE GRAPHS

AbstractLet $\Gamma $ be a graph of valency at least four whose automorphism group contains a minimally vertex-transitive subgroup G. It is proved that $\Gamma $ admits a nowhere-zero $3$ -flow if one of the following two conditions holds: (i) $\Gamma $ is of order twice an odd number and G contains a central involution; (ii) G is a direct product of a $2$ -subgroup and a subgroup of odd order.

Orthogonal root numbers of tempered parameters

We show that an orthogonal root number of a tempered L-parameter varphi decomposes as the product of two other numbers: the orthogonal root number of the principal parameter and the value on a central involution of Langlands’s central character for varphi . The formula resolves a conjecture of Gross and Reeder and computes root numbers of Weil–Deligne representations arising in a conjectural description of the Plancherel measure.

Fusion systems with Benson–Solomon components

The Benson–Solomon systems comprise an infinite ascending family of simple exotic fusion systems at the prime 2. The results we prove give significant additional evidence that these are the only simple exotic 2-fusion systems, as conjectured by Solomon. We consider a saturated fusion system F having an involution centralizer with a component C isomorphic to a Benson–Solomon fusion system, and we show under rather general hypotheses that F cannot be simple. Furthermore, we prove that if F is almost simple with these properties, then F is isomorphic to the next larger Benson–Solomon system extended by a group of field automorphisms. Our results are situated within Aschbacher’s program to provide a new proof of a major part of the classification of finite simple groups via fusion systems. One of the most important steps in this program is a proof of Walter’s Theorem for fusion systems, and our first result is specifically tailored for use in the proof of that step. We then apply Walter’s Theorem to treat the general Benson–Solomon component problem under the assumption that each component of an involution centralizer in F is on the list of currently known quasisimple 2-fusion systems.

Centralizers in pseudo-finite groups

The role of finite centralizers of involutions in pseudo-finite groups is analyzed. It is shown that a pseudo-finite group admitting a definable involutory automorphism fixing only finitely many elements is finite-by-abelian-by-finite. As a consequence, we give a model-theoretic proof of a result for periodic groups due to Hartley and Meixner. Furthermore, it is shown that any pseudo-finite group has an infinite abelian subgroup.

Finite groups with very few character values

Finite groups with very few character values are characterized. The following is the main result of this article: A finite non-abelian group has precisely four character values if and only if it is the generalized dihedral group of a non-trivial elementary abelian 3-group. The proof involves the analysis of the centralizers of involutions.

A note on involution centralizers in black box groups

Abstract Here we note a minor variation on the method in [J. N. Bray, An improved method for generating the centralizer of an involution, Arch. Math. (Basel) 74 2000, 4, 241–245] which enables calculations of C H ⁢ ( t ) {C_{H}(t)} for H a subgroup of a black box group G and t an involution of G. We then give some applications of this, the first to investigate the action of the Lyons sporadic group via conjugation on its involutions, a number of calculations in E 6 ⁢ ( 2 ) {E_{6}(2)} and E 8 ⁢ ( 2 ) {E_{8}(2)} , then finally to carry out some calculations in GL 670 ⁢ ( 2 ) {\mathrm{GL}_{670}(2)} .

A method to construct 1‐rotational factorizations of complete graphs and solutions to the oberwolfach problem

AbstractThe concept of a 1‐rotational factorization of a complete graph under a finite group was studied in detail by Buratti and Rinaldi. They found that if admits a 1‐rotational 2‐factorization, then the involutions of are pairwise conjugate. We extend their result by showing that if a finite group admits a 1‐rotational ‐factorization with even and odd, then has at most conjugacy classes containing involutions. Also, we show that if has exactly conjugacy classes containing involutions, then the product of a central involution with an involution in one conjugacy class yields an involution in a different conjugacy class. We then demonstrate a method of constructing a 1‐rotational ‐factorization under given a 1‐rotational 2‐factorization under a finite group . This construction, given a 1‐rotational solution to the Oberwolfach problem , allows us to find a solution to when the ’s are even (), and when is an odd prime, with no restrictions on the ’s.

Spinor Modules of Clifford Algebras in Classes $$N_{2k-1}$$ N 2 k - 1 and $$\Omega _{2k-1}$$ Ω 2 k - 1 are Determined by Irreducible Nonlinear Characters of Corresponding Salingaros Vee Groups

Clifford algebras $$C \ell _{2,0}$$ and $$C \ell _{1,1}$$ are isomorphic simple algebras whose Salingaros vee groups belong to a class $$N_1$$ . The algebras are isomorphic to the quotient algebra $$\mathbb {R}[D_8]/\mathcal {J}$$ of the group algebra of the dihedral group $$D_8$$ modulo an ideal $$\mathcal {J}=(1+\tau )$$ where $$\tau $$ is a central involution in $$D_8$$ . Since all irreducible characters of $$D_8$$ , including a single nonlinear character of degree 2, can be realized over $$\mathbb {R}$$ , spinor representations of the Clifford algebras can be realized over $$\mathbb {R}$$ and so $$C \ell _{2,0} \cong C \ell _{1,1} \cong \mathbb {R}(2)$$ . Spinor modules in $$C \ell _{2,0}$$ and $$C \ell _{1,1}$$ are isomorphic to irreducible $$\mathbb {R}D_8$$ -submodules of dimension 2 of the regular module $$\mathbb {R}D_8$$ . As such, they are uniquely determined by the nonlinear character of degree 2. These results are generalized to the vee groups in classes $$N_{2k-1}$$ and $$\Omega _{2k-1}$$ ( $$1 \le k \le 4$$ ). It is proven that each irreducible character of $$G_{p,q}$$ in these classes can be realized over $$\mathbb {R}.$$ Consequently, every nonlinear character of $$G_{p,q}$$ uniquely determines a spinor module of $$C \ell _{p,q}$$ which is faithful (resp. unfaithful) when $$G_{p,q}$$ is in the class $$N_{2k-1}$$ (resp. $$\Omega _{2k-1})$$ . This paper is a continuation of [1].

Reductions to simple fusion systems

We prove that if $\mathcal{E}\trianglelefteq\mathcal{F}$ are saturated fusion systems over $p$-groups $T\trianglelefteq S$, such that $C_S(\mathcal{E})\le T$, and either $Aut_{\mathcal{F}}(T)/Aut_{\mathcal{E}}(T)$ or $Out(\mathcal{E})$ is $p$-solvable, then $\mathcal{F}$ can be "reduced" to $\mathcal{E}$ by alternately taking normal subsystems of $p$-power index or of index prime to $p$. In particular, this is the case whenever $\mathcal{E}$ is simple and "tamely realized" by a known simple group. This answers a question posed by Michael Aschbacher, and is useful when analyzing involution centralizers in simple fusion systems, in connection with his program for reproving parts of the classification of finite simple groups by classifying certain 2-fusion systems.

The character table of an involution centralizer in the Dempwolff group 25·GL5(2)

The Dempwolff group D = 25· GL5(2) has an involution centralizer of the type 25· (24 :GL4(2)) of index 31 and has order 10321920. In this paper, the Fischer-Clifford matrices of the non-split extension are computed. Then the character table of is constructed using these Fischer-Clifford matrices, the ordinary character table of 24:GL4(2) and the projective character tables of the inertia factor groups H2 = 24 :(23 :GL3 (2)) and .

Heisenberg Groups and their Automorphisms over Algebras with Central Involution

Heisenberg groups over algebras with central involution and their automorphism groups are constructed. The complex quaternion group algebra over a prime field is used as an example. Its subspaces provide finite models for each of the real and complex quadratic spaces with dimension 4 or less. A model for the representations of these Heisenberg groups and automorphism groups is constructed. A pseudo-differential operator enables a parallel treatment of spaces defined over finite and real fields.

Endo-trivial modules for finite groups with Klein-four Sylow 2-subgroups

We study the finitely generated abelian group T (G) of endo-trivial kG-modules where kG is the group algebra of a finite group G over a field of characteristic $${p > 0}$$ . When the representation type of the group algebra is not wild, the group structure of T (G) is known for the cases where a Sylow p-subgroup P of G is cyclic, semi-dihedral and generalized quaternion. We investigate T (G), and more accurately, its torsion subgroup TT(G) for the case where P is a Klein-four group. More precisely, we give a necessary and sufficient condition in terms of the centralizers of involutions under which $${TT(G) = f^{-1}(X(N_{G}(P)))}$$ holds, where $${f^{-1}(X(N_{G}(P)))}$$ denotes the abelian group consisting of the kG-Green correspondents of the one-dimensional kN G (P)-modules. We show that the lift to characteristic zero of any indecomposable module in TT(G) affords an irreducible ordinary character. Furthermore, we show that the property of a module in f −1(X(N G (P))) of being endo-trivial is not intrinsic to the module itself but is decided at the level of the block to which it belongs.

A note on computing involution centralizers

For a black box group G and t an involution of G we describe a computational procedure which produces elements of CG(t) by making use of the local fusion graph F(G,X), where X is the G-conjugacy class of t.

Another Uniqueness Proof of the Dickson Simple Group G2(3)

In this article, we give a self-contained uniqueness proof for the Dickson simple group G=G2(3) using the first author's uniqueness criterion. The uniqueness proof for G2(3) was first given by Janko. His proof depends on Thompson's deep and technical characterization of G2(3). Let H′ be the amalgamated central product of SL 2(3) with itself. Then there is a unique extension H of H′ by a cyclic group of order 2 such that H has a center of order 2 and both factors SL 2(3) are normal in H. We prove that any simple group G having a 2-central involution z with centralizer CG(z)≅ H is isomorphic to G2(3).

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.

Groups saturated by finite simple groups

We describe periodic groups saturated by groups in the set ℭ = {L 2(r), r ⩾ 4; L 3(2 m ), m ⩾ 1; U 3(2 m ), m ⩾ 2; Sz(2 m ), m ⩾ 3}. As a corollary we give a description of periodic groups G saturated by finite simple groups and satisfying one of the following conditions: (a) centralizers of involutions in G are 2-closed; (b) G contains a strongly embedded 2-local subgroup.

Sebaceous gland hyperplasia of the foreskin

Two men, aged in their 20s, presented with multiple, soft, rounded papules on the prepuce. The lesions were centrally umbilicated, resembling molluscum contagiosum, but clearly distinct from Tyson's glands. Surface microscopy showed well-defined, milky-white, bag-shaped structures, which under histological examination were found to be sebaceous glands with various features of hyperplasia. A lymphocytic T-cell infiltrate, closely associated with progressive degeneration and destruction of the sebocytes, was visible around the glands. In the differential diagnosis of penile papular lesions, this unusual clinical presentation supported by dermatoscopy is consistent with preputial sebaceous gland hyperplasia. As both patients had a prominent T-cell infiltration, it is possible that under inflammatory stimulation, sebaceous glands undergo hypertrophy and gradual central involution.

Constructive membership in black-box groups

We present an algorithm to reduce the constructive membership problem for a black-box group G to three instances of the same problem for involution centralizers in G. If G is a finite simple group of Lie type in odd characteristic, then this reduction can be performed in (Monte Carlo) polynomial time.

Explicit constructions of loops with commuting inner mappings

In 2004, Csörgő constructed a loop of nilpotency class three with abelian group of inner mappings. As of now, no other examples are known. We construct many such loops from groups of nilpotency class two by replacing the product x y with x y h in certain positions, where h is a central involution. The location of the replacements is ultimately governed by a symmetric trilinear alternating form.

Centralizers of Involutions in Locally Finite Groups

The present article deals with locally finite groups G having an involution φ such that C G (φ) is an SF-group. It is shown that G possesses a normal subgroup B which is a central product of finitely many groups isomorphic to PSL(2, K i ) or SL(2, K i ) for some infinite locally finite fields K i of odd characteristic, such that [G, φ]′/B and G/[G, φ] are both SF-groups.