We generalise a technical tool, originally developed by Pervova for the study of maximal subgroups in Grigorchuk and GGS groups, to all weakly branch groups satisfying a natural condition, and in particular to all branch groups. We then use this tool to prove that every maximal subgroup of infinite index of a branch group is also a branch group. As a further application of this result, we show that every maximal subgroup of the Basilica group is of finite index.
We give a description of a finite group whose maximal subgroups possess only soluble proper subgroups, which implies the answer to the well-known question on composition factors of finite groups, whose second maximal subgroups are soluble.
In this paper, we investigate the difference between weak second maximal subgroups and second maximal subgroups. A sufficient and necessary condition is also given to describe such class of groups whose weak second maximal subgroups coincide with its second maximal subgroups(called WSM-groups) under the solvable case. As an application, we show that every non-vanishing element of a solvable WSM-group lies in its Fitting subgroup.
We prove the following results: (1) Every group is a maximal subgroup of some free idempotent generated semigroup. (2) Every finitely presented group is a maximal subgroup of some free idempotent generated semigroup arising from a finite semigroup. (3) Every group is a maximal subgroup of some free regular idempotent generated semigroup. (4) Every finite group is a maximal subgroup of some free regular idempotent generated semigroup arising from a finite regular semigroup.
We conclude from the results of Hanguang Meng and Xiuyun Guo some corollaries about the existence of strictly 2-maximal subgroups in groups. We give examples of groups that illustrate properties of strictly 2-maximal subgroups.
We establish the existence of maximal subgroups of various diferent natures in SL(n,Z). In particular, we prove that there are continuously many maximal subgroups, we provide a maximal subgroup whose action on the projective space has no dense orbits, and we produce a faithful primitive permutation representation of PSL(n,Z) which is not 2-transitive.
We classify closed abelian subgroups of the automorphism group of any compact classical simple Lie algebra whose centralizer has the same dimension as the dimension of the subgroup, and describe Weyl groups of maximal abelian subgroups.
A subgroup of a finite group G is said to be second maximal if it is maximal in every maximal subgroup of G that contains it. A question which has received considerable attention asks: can every positive integer occur as the number of the maximal subgroups that contain a given second maximal subgroup in some finite group G? Various reduction arguments are available except when G is almost simple. Following the classification of the finite simple groups, finite almost simple groups fall into three categories: alternating and symmetric groups, almost simple groups of Lie type, sporadic groups an…
Let $G$ be a finitely generated regular branch group acting by automorphisms on a regular rooted tree $T$. It is well-known that stabilizers of infinite rays in $T$ (aka parabolic subgroups) are weakly maximal subgroups in $G$, that is, maximal among subgroups of infinite index. We show that, given a finite subgroup $Q\leq G$, $G$ possesses uncountably many automorphism equivalence classes of weakly maximal subgroups containing $Q$. In particular, for Grigorchuk-Gupta-Sidki type groups this implies that they have uncountably many automorphism equivalence classes of weakly maximal subgroups tha…
We classify closed abelian subgroups of a compact simple Lie group of adjoint type and of type E having centralizer of the same dimension as the dimension of the subgroup and describe Weyl groups of maximal abelian subgroups.
The following result is received: Let $H$ be a non-normal maximal subgroup of a finite solvable group $G$ and let $q \in π(F(H/\mathrm{Core}_GH))$, then $G$ has a Sylow $q$-subgroup $Q$ such that $N_{G}(Q) \subseteq H$.
We classify abelian subgroups of the automorphism group of any compact simple Lie algebra whose centralizer has the same dimension as the dimension of the subgroup. This leads to a classification of the maximal abelian subgroups of compact simple groups of adjoint type and a classification of the fine group gradings of complex simple Lie algebras.
We determine all maximal subgroups of the almost simple groups with socle $T=\PSL(2,q)$, that is, of all groups $G$ such that $\PSL(2,q)\leqslant G\leqslant\PGammaL(2,q)$, with $q\geq 4$.
This paper is a survey on the works [MS77, MS79, MS81] on maximal subgroups in finitely generated linear groups, and the works that followed it [GG08, GG13b, GG13a, Kap03, Iva92, HO16, GM16, AGS14, Sf90, Sf98, Per05, AKT16, FG18, GS17] concerning maximal subgroups of infinite index in linear groups as well as in various other groups possessing a suitable geometry or dynamics.
In this note we give a classification of the Maximal order Abelian subgroups of finite irreducible Coxeter groups. We also prove a Weyl group analogue of Cartan's theorem that all maximal tori in a connected compact Lie group are conjugate.
Let $H$ be an open subgroup of a profinite group that can be expressed as intersection of maximal subgroups of $G.$ Given a positive real number $η,$ we say that $H$ is an $η$-intersection if there exists a family of maximal subgroups $M_1,\ldots,M_t$ such that $H=M_1\cap\ldots\cap M_t$ and $|G:M_1|\cdots|G:M_t|\le |G:H|^η$. We investigate the meaning of this property and its influence on the group structure.
Let $\mathfrak{X}$ be a class of finite groups closed under taking subgroups, homomorphic images and extensions. It is known that if $A$ is a normal subgroup of a finite group $G$ then the image of an $\mathfrak{X}$-maximal subgroup $H$ of $G$ in $G/A$ is not, in general, $\mathfrak{X}$-maximal in $G/A$. We say that the reduction $\mathfrak{X}$-theorem holds for a finite group $A$ if, for every finite group $G$ that is an extension of $A$ (i. e. contains $A$ as a normal subgroup), the number of conjugacy classes of $\mathfrak{X}$-maximal subgroups in $G$ and $G/A$ is the same. The reduction $\…
We determine all maximal subgroups of the direct product $\sc G^n$ of $\sc n$ copies of a group~$\sc G$. If $\sc G$ is finite, we show that the number of maximal subgroups of~$\sc G^n$ is a quadratic function of~$\sc n$ if $\sc G$ is perfect, but grows exponentially otherwise. We~deduce a theorem of Wiegold about the growth behaviour of the number of generators of~$\sc G^n$.
We classify closed abelian subgroups of the simple groups $G_2$, $F_4$, $Aut(so(8))$ having centralizer the same dimension as the dimension of the subgroup, as well as finite abelian subgroups of certain spin and half-spin groups having finite centralizer.
A subgroup $H$ of a group $G$ is called $\Bbb P$-{\sl subnormal} in $G$ if either $H=G$ or there is a chain of subgroups $H=H_0\subset H_1\subset...\subset H_n=G$ such that $|H_i:H_{i-1}|$ is prime for $1\le i\le n$. In this paper we study the groups all of whose 2-maximal subgroups are $\Bbb P$-subnormal.
We investigate the influence of the intersection of the $\cal F$-maximal subgroups on the structure of a finite group. In particular, answering a question of L.A Shemetkov we give conditions under which a hereditary saturated formation $\cal F$ has a property that for any finite group $G$, the ${\cal F}$-hypercentre of $G$ coincides with the intersection of all $\cal F$-maximal subgroups of $G$.
We prove the existence and uniqueness of a new maximal subgroup of the algebraic group of type $E_8$ in characteristic $3$. This has type $F_4$, and was missing from previous lists of maximal subgroups produced by Seitz and Liebeck--Seitz. We also prove a result about the finite group $H={}^3\!D_4(2)$, that if $H$ embeds in $E_8$ (in any characteristic $p$) and has two composition factors on the adjoint module then $p=3$ and $H$ lies in this new maximal $F_4$ subgroup.
A systematic study of maximal subgroups of the sporadic simple groups began in the 1960s. The work is now almost complete, only a few cases in the Monster remaining outstanding. We give a survey of results obtained, and methods used, over the past 50 years, for the classification of maximal subgroups of sporadic simple groups, and their automorphism groups.
A multi-edge spinal group is a subgroup of the automorphism group of a regular p-adic rooted tree, generated by one rooted automorphism and a finite number of directed automorphisms sharing a common directing path. We prove that torsion multi-edge spinal groups do not have maximal subgroups of infinite index. This generalizes a result of Pervova for GGS-groups.
In 1933 B.~H.~Neumann constructed uncountably many subgroups of ${\rm SL}_2(\mathbb Z)$ which act regularly on the primitive elements of $\mathbb Z^2$. As pointed out by Magnus, their images in the modular group ${\rm PSL}_2(\mathbb Z)\cong C_3*C_2$ are maximal nonparabolic subgroups, that is, maximal with respect to containing no parabolic elements. We strengthen and extend this result by giving a simple construction using planar maps to show that for all integers $p\ge 3$, $q\ge 2$ the triangle group $Γ=Δ(p,q,\infty)\cong C_p*C_q$ has uncountably many conjugacy classes of nonparabolic maxima…
Let ${\mathfrak{X}}$ be a class of finite groups closed under taking subgroups, homomorphic images, and extensions. Denote by ${\mathrm{k}}_{\mathfrak{X}}(G)$ the number of conjugacy classes ${\mathfrak{X}}$-maximal subgroups of a finite group $G$. The natural problem to describe up to conjugacy ${\mathfrak{X}}$-maximal subgroups of a given finite group is complicated by the fact that it is not inductive. In particular, generally speaking, the image of an ${\mathfrak{X}}$-maximal subgroup is not ${\mathfrak{X}}$-maximal in the image of a homomorphism. Nevertheless, there are group homomorphism…
A subgroup $Δ\leq Γ$ is commensurated if $|Δ:Δ\cap γΔγ^{-1}|<\infty$ for all $γ\in Γ$. We show a finitely generated branch group is just infinite if and only if every commensurated subgroup is either finite or of finite index. As a consequence, every commensurated subgroup of the Grigorchuk group is either finite or finite index.
A Grigorchuk-Gupta-Sidki (GGS-)group is a subgroup of the automorphism group of the $p$-adic tree for an odd prime $p$, generated by one rooted automorphism and one directed automorphism. Pervova proved that all torsion GGS-groups do not have maximal subgroups of infinite index. Here we extend the result to non-torsion GGS-groups, which include the weakly regular branch, but not branch, GGS-group.
In this paper we obtain significant bounds for the number of maximal subgroups of a given index of a finite group. These results allow us to give new bounds for the number of random generators needed to generate a finite $d$-generated group with high probability.
In this paper, we set $η(G)$ to be the number of conjugacy classes of maximal cyclic subgroups of $G$. We prove that if $G$ is a $p$-group of order $p^n$ and nilpotence class $l$, then $η(G)$ is bounded below by a linear function in $n/l$.
We prove that all maximal subgroups of the free idempotent generated semigroup over a band B are free for all B belonging to a band variety V if and only if V consists either of left seminormal bands, or of right seminormal bands.
In [Li and Chen, A new characterization of the simple group A_1(p^n), Sib. Math. J., 2012], it is proved that the simple group A_1(p^n) is uniquely determined by the set of orders of its maximal abelian subgroups. Also in [Momen and Khosravi, Groups with the same orders of maximal abelian subgroups as A_2(q), Monatsh. Math., 2013], the authors proved that if L=A_2(q), where q is not a Mersenne prime, then every finite group with the same orders of maximal abelian subgroups as L, is isomorphic to L or an extension of L by a subgroup of the outer automorphism group of L. In this paper, we prove …
We investigate the class $\mathcal{MN}$ of groups with the property that all maximal subgroups are normal. The class $\mathcal{MN}$ appeared in the framework of the study of potential counter-examples to the Andrews-Curtis conjecture. In this note we give various structural properties of groups in $\mathcal{MN}$ and present examples of groups in $\mathcal{MN}$ and not in $\mathcal{MN}$.
We use the description of the Schutzenberger automata for amalgams of finite inverse semigroups given by Cherubini, Meakin, Piochi to obtain structural results for such amalgams. Schutzenberger automata, in the case of amalgams of finite inverse semigroups, are automata with special structure possessing finite subgraphs, that contain all essential information related to the whole automaton. Using this crucial fact, and the Bass-Serre theory, we show that the maximal subgroups of an amalgamated free-product are either isomorphic to certain subgroups of the original semigroups or can be describe…
We give a structural description of the normal subgroups of subgroups of finite index in branch groups in terms of rigid stabilizers. This gives further insight into the structure lattices of branch groups introduced by the second author. We derive a condition concerning abstract commensurability of branch groups acting on the p-ary tree for any prime p.
For a finite group $G$ we investigate the difference between the maximum size MaxDim$(G)$ of an "independent" family of maximal subgroups of $G$ and maximum size $m(G)$ of an irredundant sequence of generators of $G$. We prove that MaxDim$(G)=m(G)$ if the derived subgroup of $G$ is nilpotent. However MaxDim$(G)-m(G)$ can be arbitrarily large: for any odd prime $p,$ we construct a finite soluble group with Fitting length 2 satisfying $m(G)=3$ and MaxDim$(G)=p.$
Let $G$ be a finite almost simple group. It is well known that $G$ can be generated by 3 elements, and in previous work we showed that 6 generators suffice for all maximal subgroups of $G$. In this paper we consider subgroups at the next level of the subgroup lattice - the so-called second maximal subgroups. We prove that with the possible exception of some families of rank 1 groups of Lie type, the number of generators of every second maximal subgroup of $G$ is bounded by an absolute constant. We also show that such a bound holds without any exceptions if and only if there are only finitely m…
