Products Of Finite Simple Groups Research Articles

The diameter of products of finite simple groups

Following partially a suggestion by Pyber, we prove that the diameter of a product of non-abelian finite simple groups is bounded linearly by the maximum diameter of its factors. For completeness, we include the case of abelian factors and give explicit constants in all bounds.

Coleman automorphisms of finite groups and their minimal normal subgroups

In this paper, we show that all Coleman automorphisms of a finite group with self-central minimal non-trivial characteristic subgroup are inner; therefore the normalizer property holds for these groups. Using our methods we show that the holomorph and wreath product of finite simple groups, among others, have no non-inner Coleman automorphisms. As a further application of our theorems, we provide partial answers to questions raised by M. Hertweck and W. Kimmerle. Furthermore, we characterize the Coleman automorphisms of extensions of a finite nilpotent group by a cyclic p-group. Finally, we note that class-preserving Coleman automorphisms of p-power order of some nilpotent-by-nilpotent groups are inner, extending a result by J. Hai and J. Ge, where p is a prime number.

Engel-type subgroups and length parameters of finite groups

Let $g$ be an element of a finite group $G$. For a positive integer $n$, let $E_n(g)$ be the subgroup generated by all commutators $[...[[x,g],g],\dots ,g]$ over $x\in G$, where $g$ is repeated $n$ times. By Baer's theorem, if $E_n(g)=1$, then $g$ belongs to the Fitting subgroup $F(G)$. We generalize this theorem in terms of certain length parameters of $E_n(g)$. For soluble $G$ we prove that if, for some $n$, the Fitting height of $E_n(g)$ is equal to $k$, then $g$ belongs to the $(k+1)$th Fitting subgroup $F_{k+1}(G)$. For nonsoluble $G$ the results are in terms of nonsoluble length and generalized Fitting height. The generalized Fitting height $h^*(H)$ of a finite group $H$ is the least number $h$ such that $F^*_h(H)=H$, where $F^*_0(H)=1$, and $F^*_{i+1}(H)$ is the inverse image of the generalized Fitting subgroup $F^*(H/F^*_{i}(H))$. Let $m$ be the number of prime factors of $|g|$ counting multiplicities. It is proved that if, for some $n$, the generalized Fitting height of $E_n(g)$ is equal to $k$, then $g$ belongs to $F^*_{f(k,m)}(G)$, where $f(k,m)$ depends only on $k$ and $m$. The nonsoluble length~$\lambda (H)$ of a finite group~$H$ is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if $\lambda (E_n(g))=k$, then $g$ belongs to a normal subgroup whose nonsoluble length is bounded in terms of $k$ and $m$. We also state conjectures of stronger results independent of $m$ and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups.

Embedding properties of hereditarily just infinite profinite wreath products

We study infinitely iterated wreath products of finite permutation groups w.r.t. product actions. In particular, we prove that, for every non-empty class of finite simple groups X, there exists a finitely generated hereditarily just infinite profinite group W with composition factors in X such that any countably based profinite group with composition factors in X can be embedded into W. Additionally we investigate when infinitely iterated wreath products of finite simple groups w.r.t. product actions are co-Hopfian or non-co-Hopfian.

Length-type parameters of finite groups with almost unipotent automorphisms

Let $\alpha $ be an automorphism of a finite group $G$. For a positive integer $n$, let $E_{G,n}(\alpha )$ be the subgroup generated by all commutators $[...[[x,\alpha ], \alpha ],\dots ,\alpha ]$ in the semidirect product $G\langle\alpha \rangle$ over $x\in G$, where $\alpha $ is repeated $n$ times. By Baer's theorem, if $E_{G,n}(\alpha )=1$, then the commutator subgroup $[G,\alpha ]$ is nilpotent. We generalize this theorem in terms of certain length parameters of $E_{G,n}(\alpha )$. For soluble $G$ we prove that if, for some $n$, the Fitting height of $E_{G,n}(\alpha )$ is equal to $k$, then the Fitting height of $[G,\alpha ]$ is at most $k+1$. For nonsoluble $G$ the results are in terms of the nonsoluble length and generalized Fitting height. The generalized Fitting height $h^*(H)$ of a finite group $H$ is the least number $h$ such that $F^*_h(H)=H$, where $F^*_0(H)=1$, and $F^*_{i+1}(H)$ is the inverse image of the generalized Fitting subgroup $F^*(H/F^*_{i}(H))$. Let $m$ be the number of prime factors of the order $|\alpha |$ counting multiplicities. It is proved that if, for some $n$, the generalized Fitting height of $E_{G,n}(\alpha )$ is equal to $k$, then the generalized Fitting height of $[G,\alpha ]$ is bounded in terms of $k$ and $m$. The nonsoluble length~$\lambda (H)$ of a finite group~$H$ is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if $\lambda (E_{G,n}(\alpha ))=k$, then the nonsoluble length of $[G,\alpha ]$ is bounded in terms of $k$ and $m$. We also state conjectures of stronger results independent of $m$ and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups.

Ultraproducts of quasirandom groups with small cosocles

Abstract A D-quasirandom group is a group without any non-trivial unitary representation of dimension less than D. Given a sequence of groups with increasing quasirandomness, it is natural to ask if the ultraproduct will end up with no finite dimensional unitary representation at all. This is not true in general, but we answer this question in the affirmative when the groups in question have uniform small cosocles, i.e., their quotients by small kernels are direct products of finite simple groups. Two applications of our results are given, one in triangle patterns inside quasirandom groups and one in self-bohrifying groups. Our main tools are some variations of the covering number for groups, different kinds of length functions on groups, and the classification of finite simple groups.

Hilbertian fields and Galois representations

Abstract We prove a new Hilbertianity criterion for fields in towers whose steps are Galois with Galois group either abelian or a product of finite simple groups. We then apply this criterion to fields arising from Galois representations. In particular we settle a conjecture of Jarden on abelian varieties.

On the periodic groups saturated by direct products of finite simple groups. II

On the periodic groups saturated by direct products of finite simple groups. II

On the periodic groups saturated by direct products of finite simple groups

We prove the local finiteness of a periodic group G saturated by direct products of an elementary abelian 2-group of fixed order and the simple groups L2(q) under condition that G contains an element of order 4.

A structure theorem on compact groups

We prove a new structure theorem which we call the Countable Layer Theorem. It says that for any compact group G we can construct a countable descending sequence G = Ω0(G) ⊇ … ⊇ Ωn(G) … of closed characteristic subgroups of G with two important properties, namely, that their intersection ∩∞n=1 Ωn(G) is Z0(G0), the identity component of the center of the identity component G0 of G, and that each quotient group Ωn−1(G)/Ωn(G), is a cartesian product of compact simple groups (that is, compact groups having no normal subgroups other than the singleton and the whole group).In the special case that G is totally disconnected (that is, profinite) the intersection of the sequence is trivial. Thus, even in the case that G is profinite, our theorem sharpens a theorem of Varopoulos [8], who showed in 1964 that each profinite group contains a descending sequence of closed subgroups, each normal in the preceding one, such that each quotient group is a product of finite simple groups. Our construction is functorial in a sense we will make clear in Section 1.

Infinite products of finite simple groups II

Infinite products of finite simple groups II

Infinite products of finite simple groups

We classify the sequences ⟨ S n ∣ n ∈ N ⟩ \langle S_{n} \mid n \in \mathbb {N} \rangle of finite simple nonabelian groups such that ∏ n S n \prod _{n} S_{n} has uncountable cofinality.

Graphs with Given Automorphism Group and Few Edge Orbits

A graph X is said to represent the group G with k edge (vertex) orbits if the automorphism group of X is isomorphic to G and it acts with k orbits on the set of edges (vertices, resp.) of X . It is known that (with three exceptions) every finite group can be represented with ➯2 vertex orbits (Babai, 1974). In this paper we build a framework to study the minimum number of edge orbits. We show that surprisingly large classes of groups admit a representation with a bounded number of edge-orbits. These include the groups generated by a bounded number of abelian subgroups; in particular, any direct product of finite simple groups. The case of abelian groups is related to work of Brenner, and of Gelfand and Ponomarev.

Groups which are products of finite simple groups

Groups which are products of finite simple groups

On products of finite simple groups

On products of finite simple groups

Products of finite simple groups

We call a group an R-group when it has the property that each of its simple, normal subgroups has a solvable outer-automorphism group. In this paper we study R-groups which can be written as the product of two non-Abelian simple groups. We show that if G is an R-group which is the product of two non-Abelian simple groups, then all the proper normal subgroups of G are solvable. We also show that if G is an R-group, G = AB where A, B are non-Abelian simple groups, Z(G) = 1, 1 ≠ N ≠ G, N ◁ G, and ¦ N ¦ = p k for p a prime, then the Sylow p-subgroups of both A and B are non-Abelian. This theorem restricts the possible normal subgroups of a group that is the product of simple groups. Using these and other results we determine all groups, G, which are the product of two non-Abelian simple groups, where the order of the Sylow 2-subgroup of G is less than or equal to 32. We also determine all the groups G = AB, where A, B are simple groups with Sylow 2-subgroups dihedral of order 8. We determine all groups which are the product of two non-Abelian simple groups and which have either Abelian, dihedral, or quasi-dihedral Sylow 2-subgroups.