Solvable Normal Subgroup Research Articles

On normal subgroups in automorphism groups

Abstract We describe the structure of virtually solvable normal subgroups in the automorphism group of a right-angled Artin group Aut ⁢ ( A Γ ) \mathrm{Aut}(A_{\Gamma}) . In particular, we prove that a finite normal subgroup in Aut ⁢ ( A Γ ) \mathrm{Aut}(A_{\Gamma}) has at most order two and if Γ is not a clique, then any finite normal subgroup in Aut ⁢ ( A Γ ) \mathrm{Aut}(A_{\Gamma}) is trivial. This property has implications for automatic continuity and C ∗ C^{\ast} -algebras: every algebraic epimorphism φ : L ↠ Aut ⁢ ( A Γ ) \varphi\colon L\twoheadrightarrow\mathrm{Aut}(A_{\Gamma}) from a locally compact Hausdorff group 𝐿 is continuous if and only if A Γ A_{\Gamma} is not isomorphic to Z n \mathbb{Z}^{n} for any n ≥ 1 n\geq 1 . Furthermore, if Γ is not a join and contains at least two vertices, then the set of invertible elements is dense in the reduced group C ∗ C^{\ast} -algebra of Aut ⁢ ( A Γ ) \mathrm{Aut}(A_{\Gamma}) . We obtain similar results for Aut ⁢ ( G Γ ) \mathrm{Aut}(G_{\Gamma}) , where G Γ G_{\Gamma} is a graph product of cyclic groups. Moreover, we give a description of the center of Aut ⁢ ( G Γ ) \mathrm{Aut}(G_{\Gamma}) in terms of the defining graph Γ.

Finite groups with an affine map of large order

AbstractLet 𝐺 be a group. A functionG→GG\to Gof the formx↦xα⁢gx\mapsto x^{\alpha}gfor a fixed automorphism 𝛼 of 𝐺 and a fixedg∈Gg\in Gis called anaffine map of 𝐺. The affine maps of 𝐺 form a group, called the holomorph of 𝐺. In this paper, we study finite groups 𝐺 with an affine map of large order. More precisely, we show that if 𝐺 admits an affine map of order larger than12⁢|G|\frac{1}{2}\lvert G\rvert, then 𝐺 is solvable of derived length at most 3. We also show that, more generally, for eachρ∈(0,1]\rho\in(0,1], if 𝐺 admits an affine map of order at leastρ⁢|G|\rho\lvert G\rvert, then the largest solvable normal subgroup of 𝐺 has derived length at most4⁢⌊log2⁡(ρ−1)⌋+34\lfloor\log_{2}(\rho^{-1})\rfloor+3.

Algebraic lattices of solvably saturated formations and their applications

In each group G, we select a system of subgroups $$\tau (G)$$ and say that $$\tau$$ is a subgroup functor if $$G \in \tau (G)$$ for every group G, and for every epimorphism $$\varphi : A \rightarrow B$$ and any $$H \in \tau (A)$$ and $$T \in \tau (B)$$ , we have $$H^\varphi \in \tau (B)$$ and $$T^{\varphi ^{-1}} \in \tau (A)$$ . We consider only subgroup functors $$\tau$$ such that for any group G all subgroups of $$\tau (G)$$ are subnormal in G. For any set of groups $${\mathfrak {X}}$$ , the symbol $${s}_\tau ({\mathfrak {X}})$$ denotes the set of groups H such that $$H \in \tau (G)$$ for some group $$G \in {\mathfrak {X}}$$ . A formation $${\mathfrak {F}}$$ is $$\tau$$ -closed if $${s}_\tau ({\mathfrak {F}}) = {\mathfrak {F}}$$ . The Frattini subgroup $${\varPhi }(G)$$ of a group G is the intersection of all maximal subgroups of G. A formation $${\mathfrak {F}}$$ is said to be solvably saturated if it contains each group G with $$G/{\varPhi }(N) \in {\mathfrak {F}}$$ for some solvable normal subgroup N of G. Composition formations are precisely solvably saturated formations. It is shown that the lattice of all $$\tau$$ -closed totally composition formations is algebraic.

On the number of irreducible real-valued characters of a finite group

We prove that there exists an integer-valued function f on positive integers such that if a finite group G has at most k real-valued irreducible characters, then |G/Sol(G)|≤f(k), where Sol(G) denotes the largest solvable normal subgroup of G. In the case k=5, we further classify G/Sol(G). This partly answers a question of Iwasaki [15] on the relationship between the structure of a finite group and its number of real-valued irreducible characters.

Solvable normal subgroups of 2-knot groups

If [Formula: see text] is an orientable, strongly minimal [Formula: see text]-complex and [Formula: see text] has one end, then it has no nontrivial locally finite normal subgroup. Hence, if [Formula: see text] is a 2-knot group, then (a) if [Formula: see text] is virtually solvable, then either [Formula: see text] has two ends or [Formula: see text], with presentation [Formula: see text], or [Formula: see text] is torsion-free and polycyclic of Hirsch length 4 (b) either [Formula: see text] has two ends, or [Formula: see text] has one end and the center [Formula: see text] is torsion-free, or [Formula: see text] has infinitely many ends and [Formula: see text] is finite, and (c) the Hirsch–Plotkin radical [Formula: see text] is nilpotent.

Algorithms for Group Isomorphism via Group Extensions and Cohomology

The isomorphism problem for finite groups of order $n$ (GpI) has long been known to be solvable in $n^{\log n+O(1)}$ time, but only recently were polynomial-time algorithms designed for several interesting group classes. Inspired by recent progress, we revisit the strategy for GpI via the extension theory of groups. The extension theory describes how a normal subgroup $N$ is related to $G/N$ via $G$, and this naturally leads to a divide-and-conquer strategy that “splits” GpI into two subproblems: one regarding group actions on other groups, and one regarding group cohomology. When the normal subgroup $N$ is abelian, this strategy is well known. Our first contribution is to extend this strategy to handle the case when $N$ is not necessarily abelian. This allows us to provide a unified explanation of all recent polynomial-time algorithms for special group classes. Guided by this strategy, to make further progress on GpI, we consider central-radical groups, proposed in Babai et al. [Code equivalence and group isomorphism, in Proceedings of the 22nd Annual ACM--SIAM Symposium on Discrete Algorithms (SODA'11), SIAM, Philadelphia, 2011, ACM, New York, pp. 1395--1408]: the class of groups such that $G$ modulo its center has no abelian normal subgroups. This class is a natural extension of the group class considered by Babai et al. [Polynomial-time isomorphism test for groups with no abelian normal subgroups (extended abstract), in International Colloquium on Automata, Languages, and Programming (ICALP), 2012, pp. 51--62], namely those groups with no abelian normal subgroups. Following the above strategy, we solve GpI in $n^{O(\log \log n)}$ time for central-radical groups, and in polynomial time for several prominent subclasses of central-radical groups. We also solve GpI in $n^{O(\log\log n)}$ time for groups whose solvable normal subgroups are elementary abelian but not necessarily central. As far as we are aware, this is the first time there have been worst-case guarantees on an $n^{o(\log n)}$-time algorithm that tackles both aspects of GpI---actions and cohomology---simultaneously. Prior to this work, the best proven upper bounds on algorithms for groups with central radicals were $n^{O(\log n)}$, even for groups with a central radical of constant size, such as ${Rad}(G) = Z(G)=\mathbb{Z}_2$. To develop our new algorithms we utilize several mathematical results on the detailed structure of cohomology classes, as well as algorithmic results for code equivalence, coset intersection, and cyclicity testing of modules over finite-dimensional associative algebras. We also suggest several promising directions for future work.

On weakly ℨ-permutable subgroups of finite groups

Let ℨ be a complete set of Sylow subgroups of a finite group G, that is, for each prime p dividing the order of G, ℨ contains exactly one and only one Sylow p-subgroup of G. A subgroup H of G is said to be ℨ-permutable of G if H permutes with every member of ℨ. A subgroup H of G is said to be a weakly ℨ-permutable subgroup of G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K ≤ Hℨ, where Hℨ is the subgroup of H generated by all those subgroups of H which are ℨ-permutable subgroups of G. In this paper, we prove that if p is the smallest prime dividing the order of G and the maximal subgroups of Gp ∈ ℨ are weakly ℨ-permutable subgroups of G, then G is p-nilpotent. Moreover, we prove that if 𝔉 is a saturated formation containing the class of all supersolvable groups, then G ∈ 𝔉 iff there is a solvable normal subgroup H in G such that G/H ∈ 𝔉 and the maximal subgroups of the Sylow subgroups of the Fitting subgroup F(H) are weakly ℨ-permutable subgroups of G. These two results generalize and unify several results in the literature.

On a graph of a $p$-solvable normal subgroup

On a graph of a $p$-solvable normal subgroup

Character kernels and degree ratios in finite groups

Let K = ker ( χ ) be the kernel of an irreducible character χ of a finite group G, and let S be the largest solvable normal subgroup of K. We show that if the degree of χ is large compared to the degrees of other irreducible characters of G, or if the kernel K is small compared to the kernels of other irreducible characters of G, then the Fitting height of S is small. Also, we show that the derived length of a nonabelian solvable group is bounded by a logarithmic function of the ratio b / c , where b is the largest irreducible character degree of G and c is the smallest nonlinear irreducible character degree. Finally, using the classification of simple groups, we show that certain kernels of irreducible characters of large degree must be solvable.

A characterization of the full wreath product

A groupoid [3, 17] is a set Q endowed with a binary product, that is, a map from Q × Q to Q. In his 1964 paper [7], Bernd Fischer studied distributive quasigroups, which by definition are groupoids Q for which right multiplication by any fixed element gives an automorphism of Q as does left multiplication. Fischer proved that the right multiplication group R(Q) of a finite distributive quasigroup Q is solvable. He did this by showing that, for a minimal counterexample, the right multiplications T = {μa : g 7→ ga | a ∈ Q } are a generating conjugacy class of involutions in R(Q) ≤ Aut(Q) with the additional property that |tr| = 3 for distinct t and r from T . He then proved that this property forces finite R(Q) to have a normal 3-group of index 2. This led Fischer to consider [8, 9, 10] the extent to which finite symmetric groups can be characterized through being generated by a conjugacy class of involutions with all products of order 1, 2, or 3—a class of 3-transpositions, since the model is the transposition (2-cycle) class of Sym(Ω), the symmetric group on the set Ω. In a landmark theorem [10], Fischer found all finite 3transposition groups with no nontrivial solvable normal subgroups, discovering three new sporadic simple groups along the way. At the same time that Fischer was considering distributive quasigroups, George Glauberman was working on certain special groupoids, called Bruck loops. Glauberman [13] proved that finite Bruck and finite Moufang loops of odd order are solvable. His approach was similar to Fischer’s. He constructed a canonical conjugacy class T of involutory loop permutations with the additional property that |tr| was always odd for t and r from T . In his famous Z∗-theorem [14], Glauberman then proved that a finite group generated by such a class T has a normal subgroup of odd order and index 2 (a result also proved by Fischer [7] in the special case where all orders |tr| are powers of some fixed odd prime). Fischer’s and Glauberman’s work on finite quasigroups and loops had a profound effect on the theory of finite simple groups. For a normal set of involutions

On groups and semigroups of matrices with spectra in a finitely generated field

Let K be an algebraically closed field of characteristic zero and F<K a subfield which is finitely generated over the prime field. Assume is a semigroup of invertible matrices such that the spectra of all the elements of are contained in F. Then the group , generated by , contains a solvable normal subgroup of finite index. As a consequence, it follows that an irreducible semigroup such that the spectra of all the elements of are contained in F is conjugate to a subsemigroup of Mn (F).

Asymptotic properties of convolution operators and limits of triangular arrays on locally compact groups

We consider a locally compact group G and a limiting measure μ of a commutative infinitesimal triangular system (c.i.t.s.) A of probability measures on G. We show, under some restrictions on G, μ or A, that μ belongs to a continuous one-parameter convolution semigroup. In particular, this result is valid for symmetric c.i.t.s. A on any locally compact group G. It is also valid for a limiting measure μ which has 'full' support on a Zariski connected F-algebraic group G, where F is a local field, and any one of the following conditions is satisfied: (1) G is a compact extension of a closed solvable normal subgroup, in particular, G is amenable, (2) μ has finite one-moment or (3) μ has density and in case the characteristic of F is positive, the radical of G is F-defined. We also discuss the spectral radius of the convolution operator of a probability measure on a locally compact group G, we show that it is always positive for any probability measure on G, and it is also multiplicative in case of symmetric commuting measures.

On Two Theorems of Finite Solvable Groups

For a finite group G, let τ(G) denote a set of primes such that a prime p belongs to τ(G) if and only if p is a divisor of the index of some maximal subgroup of G. It is proved that if G satisfies any one of the following conditions: (1) G has a p-complement for each p∈τ(G); (2) ∣τ(G)∣=2; (3) the normalizer of a Sylow p-subgroup of G has prime power index for each odd prime p∈τ(G); then G either is solvable or G/Sol(G)≅(2, 7) where Sol(G) is the largest solvable normal subgroup of (G).

Affine Rank 3 Groups on Symmetric Designs

In this note we determine the symmetric designs which admit a primitive rank 3 group with a solvable normal subgroup. This completes the investigations of Dempwolff [4].

On the Prime Radical of PI-Representable Groups

The notion of PI-representable groups is introduced; these are subgroups of invertible elements of a PI-algebra over a field. It is shown that a PI-representable group has a largest locally solvable normal subgroup, and this subgroup coincides with the prime radical of the group. The prime radical of a finitely generated PI-representable group is solvable. The class of PI-representable groups is a generalization of the class of linear groups because in the groups of the former class the largest locally solvable normal subgroup can be not solvable.

On Solvable Normal Subgroups of Finite Groups

We consider solvable invariant subgroups of a finite group with bounded primary indices of maximal subgroups. We establish that an invariant subgroup of this type belongs to the product of classical formations and investigate its dispersibility.

Representing Subgroups of Finitely Presented Groups by Quotient Subgroups

This article proposes to represent subgroups of finitely presented groups by their image in a quotient. It gives algorithms for basic operations in this representation and investigates how iteration of this approach can be used to extend known quotients groups with a solvable normal subgroup.

The central limit problem on locally compact groups

It is shown that the limit μ of a commutative infinitesimal triangular system Δ on a totally disconnected locally compact groupG is embeddable in a continuous one-parameter convolution semigroup if either (1)G is a compact extension of a closed solvable normal subgroup or (2)G is discrete and Δ is normal or (3)G is a discrete linear group over a field of characteristic zero. For a special triangular system of convolution powers\(\left( {\mu _\nu ^{\kappa _\nu } \to \mu ,\mu _n \to \delta _3 } \right)\), the above is shown to hold without any of the conditions (1)–(3). For a general locally compact groupG necessary conditions are obtained for the embeddability of a shift of limit μ of Δ; in particular, the conditions are trivially satisfied whenG is abelian. Also, the embedding of a limit of a symmetric system onG is shown to hold under condition (1) as above.

The fundamental group of a $\mathcal{CP}^2$ complement of a branch curve as an extension of a solvable group by a symmetric group

The main result in this paper is as follows: Theorem. Let S be the branch curve in CP 2 of a generic projection of a Veronese surface. Then �1(CP 2 S) is an extension of a solvable group by a symmetric group. A group with the property mentioned in the theorem is "almost solvable" in the sense that it contains a solvable normal subgroup of finite index. We pose the following question. Question. For which families of simply connected algebraic surfaces of general type is the fundamental group of the complement of the branch curve of a generic projec- tion to CP 2 an extension of a solvable group by a symmetric group?

Groups without Faithful Transitive Permutation Representations of Small Degree

A subgroupHof a groupGiscore-freeifHcontains no non-trivial normal subgroup ofG, or equivalently the transitive permutation representation ofGon the cosets ofHis faithful. We study the obstacles to a group having large core-free subgroups. We call a subgroupDa “dedekind” subgroup ofGif all subgroups ofDare normal inG. Our main result is the following: If a finite groupGhas no core-free subgroups of order greater thank, thenGhas two dedekind subgroupsD1andD2such that every subgroup inGof order greater thanf(k) has non-trivial intersection with eitherD1orD2(wherefis a fixed function independent ofG). Examples show that the dedekind subgroups need not have index bounded by a function ofk, and the result would not be true with one dedekind subgroup instead of two. We exhibit various related properties ofp-groups and infinite locally finite groups without large core-free subgroups, including the following: IfGis a locally finite group with no infinite core-free subgroup, then every infinite subgroup ofGcontains a non-trivial cyclic normal subgroup ofG. We also exhibit asymptotic bounds for some related problems, including the following: If a groupGhas a solvable subgroup of indexn, thenGhas a solvable normal subgroup of index at mostnc(for some absolute constantc). IfGis a transitive permutation group of degreenwith cyclic point-stabilizer subgroup, then |G|<n7.