Abelian Normal Subgroup Research Articles

Sharply 3-transitive groups

We construct the first sharply 3-transitive groups not arising from a near field, i.e. point stabilizers have no nontrivial abelian normal subgroup.

TTHE CONJUGACY CLASSES OF METABELIAN GROUPS OF ORDER AT MOST 24

In this paper, G denotes a non-abelian metabelian group and denotes conjugacy class of the element x in G. Conjugacy class is an equivalence relation and it partitions the group into disjoint equivalence classes or sets. Meanwhile, a group is called metabelian if it has an abelian normal subgroup in which the factor group is also abelian. It has been proven by an earlier researcher that there are 25 non-abelian metabelian groups of order less than 24 which are considered in this paper. In this study, the number of conjugacy classes of non-abelian metabelian groups of order less than 24 is computed.

On Finite Groups with an Irreducible Character Large Degree

Let G be a finite nontrivial group with an irreducible complex character χ of degree d = χ(1). It is known from the orthogonality relation that the sum of the squares of degrees of irreducible characters of G is equal to the order of G. N. Snyder proved that if |G| = d(d + e), then the order of G is bounded in terms of e, provided e > 1. Y. Berkovich proved that in the case e = 1 the group G is Frobenius with the complement of order d. We study a finite nontrivial group G with an irreducible complex character Θ such that |G| ≤ 2Θ(1)2 and Θ(1) = pq, where p and q are different primes. In this case we prove that G is solvable groups with abelian normal subgroup K of index pq. We use the classification of finite simple groups and prove that the simple nonabelian group whose order is divisible by a prime p and of order less than 2p4 is isomorphic to L2(q), L3(q), U3(q), Sz(8), A7, M11 or J1.

Semiregular Automorphisms of Cubic Vertex-Transitive Graphs and the Abelian Normal Quotient Method

We characterise connected cubic graphs admitting a vertex-transitive group of automorphisms with an abelian normal subgroup that is not semiregular. We illustrate the utility of this result by using it to prove that the order of a semiregular subgroup of maximum order in a vertex-transitive group of automorphisms of a connected cubic graph grows with the order of the graph.

Residually finite algorithmically finite groups, their subgroups and direct products

We construct an infinite finitely generated recursively presented residually finite algorithmically finite group $G$ answering thereby a question of Myasnikov and Osin. Moreover, $G$ is "very infinite" and "very algorithmically finite" in the sense that $G$ contains an infinite abelian normal subgroup while all finite Cartesian powers of $G$ are algorithmically finite (i.e., for any positive integer $n$, there is no algorithm which writes out an infinite sequence of pairwise different elements of $G^n$). We also state several related problems.

Fitting Height of a Finite Group with a Metabelian Group of Automorphisms

Let M = FH be a finite group that is a product of a normal abelian subgroup F and an abelian subgroup H. Assume that all elements in M∖F have prime order p, and F has at most one subgroup of order p. Examples of such groups are dihedral groups for p = 2 and the semidirect product of a cyclic group F by a group H of prime order p such that C F (H) = 1 or |C F (H)| =p and C F/C F (H)(H) = 1. Suppose that M acts on a finite group G in such a manner that C G (F) = 1. We prove that the Fitting height h(G) of G is at most h(C G (H))+ 1. Moreover, the Fitting series of C G (H) coincides with the intersection of C G (H) with the Fitting series of G.

On the Chern‐Ricci flow and its solitons for Lie groups

This paper is concerned with Chern‐Ricci flow evolution of left‐invariant hermitian structures on Lie groups. We study the behavior of a solution, as t is approaching the first time singularity, by rescaling in order to prevent collapsing and obtain convergence in the pointed (or Cheeger‐Gromov) sense to a Chern‐Ricci soliton. We give some results on the Chern‐Ricci form and the Lie group structure of the pointed limit in terms of the starting hermitian metric and, as an application, we obtain a complete picture for the class of solvable Lie groups having a codimension one normal abelian subgroup. We have also found a Chern‐Ricci soliton hermitian metric on most of the complex surfaces which are solvmanifolds, including an unexpected shrinking soliton example.

On finite subgroups of the classical groups

Abstract In 1878, Jordan showed that a finite subgroup of GL ( n , ℂ ) ${\operatorname{GL}(n,\mathbb {C})}$ must possess an abelian normal subgroup whose index is bounded by a function of n alone. In previous papers, the author obtained optimal bounds; in particular, a generic bound ( n + 1 ) ! ${(n+1)!}$ was obtained when n ≥ 71 ${n\ge 71}$ , achieved by the symmetric group S n+1. In this paper, analogous bounds are obtained for the finite subgroups of the complex symplectic and orthogonal groups. In the case of Sp ( 2 n , ℂ ) ${\operatorname{Sp}(2n,\mathbb {C})}$ the optimal bound is ( 60 ) n · n ! ${(60)^{n}\cdot n!}$ , achieved by the wreath product SL 2 ( 5 ) wr S n ${\operatorname{SL}_{2}(5)\operatorname{wr}S_{n}}$ acting naturally on the direct sum of n 2-dimensional spaces; for the orthogonal groups O ( n , ℂ ) ${\mathrm {O}(n,\mathbb {C})}$ , the generic linear group bound of ( n + 1 ) ! ${(n+1)!}$ is achieved as soon as n ≥ 25 ${n\ge 25}$ .

A note on groups generated by involutions and sharply 2-transitive groups

LetGGbe a group generated by a setCCof involutions which is closed under conjugation. Letπ\pibe a set of odd primes. Assume that either (1)GGis solvable, or (2)GGis a linear group.We show that if the product of any two involutions inCCis aπ\pi-element, thenGGis solvable in both cases andG=Oπ(G)⟨t⟩G=O_{\pi }(G)\langle t\rangle, wheret∈Ct\in C.If (2) holds and the product of any two involutions inCCis a unipotent element, thenGGis solvable.Finally we deduce that ifG\mathcal {G}is a sharply22-transitive (infinite) group of odd (permutational) characteristic, such that every33involutions inG\mathcal {G}generate a solvable or a linear group; or ifG\mathcal {G}is linear of (permutational) characteristic0,0,thenG\mathcal {G}contains a regular normal abelian subgroup.

Generalized Quadrangles and Transitive Pseudo‐Hyperovals

AbstractA pseudo‐hyperoval of a projective space , q even, is a set of subspaces of dimension such that any three span the whole space. We prove that a pseudo‐hyperoval with an irreducible transitive stabilizer is elementary. We then deduce from this result a classification of the thick generalized quadrangles that admit a point‐primitive, line‐transitive automorphism group with a point‐regular abelian normal subgroup. Specifically, we show that is flag‐transitive and isomorphic to , where is either the regular hyperoval of PG(2, 4) or the Lunelli–Sce hyperoval of PG(2, 16).

Noether’s problem for abelian extensions of cyclicp-groups

Let $K$ be a field and $G$ be a finite group. Let $G$ act on the rational function field $K(x(g):g\in G)$ by $K$ automorphisms defined by $g\cdot x(h)=x(gh)$ for any $g,h\in G$. Denote by $K(G)$ the fixed field $K(x(g):g\in G)^G$. Noether's problem then asks whether $K(G)$ is rational (i.e., purely transcendental) over $K$. The first main result of this article is that $K(G)$ is rational over $K$ for a certain class of $p$-groups having an abelian subgoup of index $p$. The second main result is that $K(G)$ is rational over $K$ for any group of order $p^5$ or $p^6$ ($p$ is an odd prime) having an abelian normal subgroup such that its quotient group is cyclic. (In both theorems we assume that if $char K\ne p$ then $K$ contains a primitive $p^e$-th root of unity, where $p^e$ is the exponent of $G$.)

On Jordan type bounds for finite groups acting on compact 3-manifolds

By a classical result of Jordan, each finite subgroup of a complex linear group \({{\rm GL}_n(\mathbb{C})}\) has an abelian normal subgroup whose index is bounded by a constant depending only on n. It has been asked whether this remains true for finite subgroups of the diffeomorphism group Diff(M) of every compact manifold M; in the present paper, using the geometrization of 3-manifolds, we prove it for compact 3-manifolds M.

Notes on the BMS group in three dimensions: I. Induced representations

The Bondi-Metzner-Sachs group in three dimensions is the symmetry group of asymptotically flat three-dimensional spacetimes. It is the semi-direct product of the diffeomorphism group of the circle with the space of its adjoint representation, embedded as an abelian normal subgroup. The structure of the group suggests to study induced representations; we show here that they are associated with the well-known coadjoint orbits of the Virasoro group and provide explicit representations in terms of one-particle states.

Determinateness of Torsion-Free Abelian Groups by Their Holomorphs and Almost Holomorphic Isomorphism

In this paper, we study holomorphically isomorphic Abelian groups, i.e., Abelian groups with isomorphic holomorphs. We also study a generalization of the concept of holomorphic isomorphism, namely, almost holomorphic isomorphism, which is deeply connected with normal Abelian subgroups of holomorphs of Abelian groups. Torsion-free Abelian groups that are determined by their holomorphs are highlighted from different classes. In particular, it has been found that any homogeneous separable group can be determined by its holomorph in the class of all Abelian groups.

Bounding group orders by large character degrees: A question of Snyder

Abstract Let G be a nonabelian finite group and let d be an irreducible character degree of G. Then there is a positive integer e so that |G| = d(d + e). Snyder has shown that if e > 1, then |G| is bounded by a function of e. This bound has been improved by Isaacs and by Durfee and Jensen. In this paper we will show, for groups having a nontrivial, abelian normal subgroup, that |G| ≤ e 4 - e 3. Given that there are a number of solvable groups that meet this bound, it is best possible. Our work makes use of results regarding Camina pairs, Gagola characters, and Suzuki 2-groups.

Degenerate group of type A: Representations and flag varieties

We consider the degeneration of a simple Lie group which is a semidirect product of its Borel subgroup and a normal Abelian unipotent subgroup. We introduce a class of highest weight representations of the degenerate group of type A, generalizing the construction of PBW-graded representations of the classical group (PBW is an abbreviation for “Poincare-Birkhoff-Witt”). Following the classical construction of flag varieties, we consider the closures of orbits of the Abelian unipotent subgroup in projectivizations of the representations. We show that the degenerate flag varieties Fna and their desingularizations Rn can be obtained via this construction. We prove that the coordinate ring of Rn is isomorphic as a vector space to the direct sum of the duals of the highest weight representations of the degenerate group. At the end we state several conjectures on the structure of the highest weight representations of the degenerate group of type A.

ON IRREDUCIBILITY OF INDUCED MODULES AND AN ADAPTATION OF THE WIGNER-MACKEY METHOD OF LITTLE GROUPS

This paper deals with sufficiency conditions for irreducibility of certain induced modules. We also construct irreducible representations for a group G over a field <TEX>$\mathbb{K}$</TEX> where the group G is a semidirect product of a normal abelian subgroup N and a subgroup H. The main results are proved with the assumption that char <TEX>$\mathbb{K}$</TEX> does not divide |G| but there is no assumption made of <TEX>$\mathbb{K}$</TEX> being algebraically closed.

Bogomolov multipliers and retract rationality for semidirect products

Let G be a finite group. The Bogomolov multiplier B0(G) is constructed as an obstruction to the rationality of C(V)G where G→GL(V) is a faithful representation over C. We prove that, for any finite groups G1 and G2, B0(G1×G2)→∼B0(G1)×B0(G2) under the restriction map. If G=N⋊G0 with gcd{|N|,|G0|}=1, then B0(G)→∼B0(N)G0×B0(G0) under the restriction map. For any integer n, we show that there are non-direct product p-groups G1 and G2 such that B0(G1) and B0(G2) contain subgroups isomorphic to (Z/pZ)n and Z/pnZ respectively. On the other hand, if k is an infinite field and G=N⋊G0 where N is an abelian normal subgroup of exponent e satisfying that ζe∈k, we will prove that, if k(G0) is retract k-rational, then k(G) is also retract k-rational provided that certain “local” conditions are satisfied; this result generalizes previous results of Saltman and Jambor [18].

Arc transitive distance regular covers of cliques

Abstract It is consider edge-symmetric distance-regular graph Γ with intersection array { k , ( r − 1 ) μ , 1 ; 1 , μ , k } . Let G = Aut ( Γ ) . Then G acts two-transitively on the set Σ of antipodal classes (fibres) of Γ. This graphs are classified in the next cases: 1) r = 2 or r = k , 2) λ = μ , 3) G contains abelian normal subgroup N regular on Σ (affine case).

On the commuting probability and supersolvability of finite groups

For a finite group $$G$$ , let $$d(G)$$ denote the probability that a randomly chosen pair of elements of $$G$$ commute. We prove that if $$d(G)>1/s$$ for some integer $$s>1$$ and $$G$$ splits over an abelian normal nontrivial subgroup $$N$$ , then $$G$$ has a nontrivial conjugacy class inside $$N$$ of size at most $$s-1$$ . We also extend two results of Barry, MacHale, and Ni She on the commuting probability in connection with supersolvability of finite groups. In particular, we prove that if $$d(G)>5/16$$ then either $$G$$ is supersolvable, or $$G$$ isoclinic to $$A_4$$ , or $$G/\mathbf{Z}(G)$$ is isoclinic to $$A_4$$ .