We describe the structure of Sylow {\ell}-subgroups of a finite reduc-tive group G(Fq) when q $\not\equiv$ 0 (mod {\ell}) that we find governed by a complex reflection group attached to G and {\ell}, which depends on {\ell} only through the set of cyclotomic factors of the generic order of G(Fq) whose value at q is divisible by {\ell}. We also tackle the more general case G F where F is an isogeny whose a power is a Frobenius morphism.
A subgroup $H$ of a finite group $G$ is called submodular in $G$, if we can connect $H$ with $G$ by a chain of subgroups, each of which is modular (in the sense of Kurosh) in the next. If a group $G$ is supersoluble and every Sylow subgroup of $G$ is submodular in $G$, then $G$ is called strongly supersoluble. The properties of groups with submodular Sylow subgroups are obtained. In particular, we proved that in a group every Sylow subgroup is submodular if and only if the group is Ore dispersive and every its biprimary subgroup is strongly supersoluble.
The size of minimal generating set for commutator of Sylow 2-subgroup of alternating group was found. Given a permutational wreath product of finite cyclic groups sequence we prove that the commutator width of such groups is 1 and we research some properties of its commutator subgroup. It was shown that $(Syl_2 A_{2^k})^2 = Syl'_2 (A_{2^k}), \, k>2$. A new approach to presentation of Sylow 2-subgroups of alternating group ${A_{2^{k}}}$ was applied. As a result the short proof that the commutator width of Sylow 2-subgroups of alternating group ${A_{2^{k}}}$, permutation group ${S_{2^{k}}}$ and …
Let $G$ be a finite solvable group. We show that $G$ does not have a normal nonabelian Sylow $p$-subgroup when its prime character degree graph $Δ(G)$ satisfies a technical hypothesis.
A subgroup $A$ of a group~$G$ is said to be {\sl NS-supplemented} in $G$, if there exists a subgroup~$B$ of $G$ such that $G=AB$ and whenever $X$~is a normal subgroup of~$A$ and $p\in π(B)$, there exists a Sylow $p$-subgroup~$B_p$ of~$B$ such that $XB_p=B_pX$. In this paper, we proved the supersolubility of a group with NS-supplemented non-cyclic Sylow subgroups. The solubility of a group with NS-supplemented maximal subgroups is obtained.
Slattery (2007) described computational methods to enumerate, construct, and identify finite groups of squarefree order. We generalise Slattery's result to the class of finite groups that have cyclic Sylow subgroups and provide an implementation for the computer algebra system GAP.
The Sylow p-subgroups of the symmetric group S_p^n satisfy the appropriate generalization of Maschke's Theorem to the case of a p'-group acting on a (not necessarily abelian) p-group. Moreover, some known results about the Sylow p-subgroups of S_p^n are stated in a form that is true for all primes p.
An explicit and elementary proof is given to the fact that Suzuki and Ree groups can be decomposed into the product of 4 of their Sylow p-subgroups, where p is the defining characterictic.
We determine, for $p$ odd, all saturated fusion systems on a Sylow $p$-subgroup $S$ of the unitary group $SU_4(p)$ and we prove that they are all realizable by finite groups. In particular, we prove that $S$ does not support any exotic fusion systems.
Let $G$ be a finite simple group of Lie type and let $P$ be a Sylow $2$-subgroup of $G$. In this paper, we prove that for any nontrivial element $x \in G$, there exists $g \in G$ such that $G = \langle P, x^g \rangle$. By combining this result with recent work of Breuer and Guralnick, we deduce that if $G$ is a finite nonabelian simple group and $r$ is any prime divisor of $|G|$, then $G$ is generated by a Sylow $2$-subgroup and a Sylow $r$-subgroup.
For any prime $p$ and $S$ a $p$-group isomorphic to a Sylow $p$-subgroup of $\mathrm{G}_2(p^n)$ or $\mathrm{PSU}_4(p^n)$ with $n\in\mathbb{N}$, we determine all saturated fusion systems supported on $S$ up to isomorphism.
Let $σ=\{σ_{i}|i\in I\}$ be a partition of the set of all primes $\mathbb{P}$, $G$ a finite group and $σ(G)=\{σ_{i}|σ_{i}\cap π(|G|)\neq\emptyset\}$. A subgroup $S$ of a group $G$ is called a $σ_i$-sylowizer of a $σ_i$-subgroup $R$ in $G$ if $S$ is maximal in $G$ with respect to having $R$ as its Hall $σ_i$-subgroup. The main aim of this paper is to investigate the influence of $σ_i$-sylowizers on the structure of finite groups. We obtained some new characterizations of supersoluble groups by the permutability of the $σ_i$-sylowizers of some $σ_i$-subgroups.
Assume that all the Sylow subgroups of a finite group $G$ can be generated by $d$ elements. Then the expected number of elements of $G$ which have to be drawn at random, with replacement, before a set of generators is found, is at most $d+η$ with $η\sim 2.875065.$
In this article the investigation of Sylows p-subgroups of ${{A}_{n}}$ and ${{S}_{n}}$, which was started in article of U. Dmitruk, V. Suschansky "Structure of 2-sylow subgroup of symmetric and alternating group" and article of R.~Skuratovskii "Corepresentation of a Sylow p-subgroup of a group S_n" \cite{Dm, Sk, Paw} is continued. Let $Syl_2{A_{2^k}}$ and $Syl_2{A_{n}}$ be Sylow 2-subgroups of corresponding alternating groups $A_{2^k}$ and $A_{n}$. We find a least generating set and a structure for such subgroups $Sy{{l}_{2}}{{A}_{{{2}^{k}}}}$ and $Syl_2{A_{n}}$ and commutator width of $Syl_2{…
Let $T$ be a finite simple group of Lie type in characteristic $p$, and let $S$ be a Sylow subgroup of $T$ with maximal order. It is well known that $S$ is a Sylow $p$-subgroup except in an explicit list of exceptions, and that $S$ is always `large' in the sense that $|T|^{1/3} < |S| \leqslant |T|^{1/2}$. One might anticipate that, moreover, the Sylow $r$-subgroups of $T$ with $r \neq p$ are usually significantly smaller than $S$. We verify this hypothesis by proving that for every $T$ and every prime divisor $r$ of $|T|$ with $r \neq p$, the order of the Sylow $r$-subgroup of $T$ at most $|T|…
We say that a finite group $G$ acting on a set $Ω$ has Property $(*)_p$ for a prime $p$ if $P_ω$ is a Sylow $p$-subgroup of $G_ω$ for all $ω\inΩ$ and Sylow $p$-subgroups $P$ of $G$. Property $(*)_p$ arose in the recent work of Tornier (2018) on local Sylow $p$-subgroups of Burger-Mozes groups, and he determined the values of $p$ for which the alternating group $A_n$ and symmetric group $S_n$ acting on $n$ points has Property $(*)_p$. In this paper, we extend this result to finite $2$-transitive groups and we give a structural characterisation result for the finite primitive groups that satisfy…
Given a permutational wreath product sequence of cyclic groups of prime order we research a commutator width of such groups and some properties of its commutator subgroup. Commutator width of Sylow 2-subgroups of alternating group $A_{2^{k}}$, permutation group $S_{2^k}$ and $C_p \wr B$ were founded. The result of research was extended on subgroups $(Syl_2 {A_{2^k}})'$, $p>2$. The paper presents a construction of commutator subgroup of Sylow 2-subgroups of symmetric and alternating groups. Also minimal generic sets of Sylow 2-subgroups of $A_{2^k}$ were founded. Elements presentation of $(Syl_…
For $S$ a Sylow $p$-subgroup of the group $\mathrm{G}_2(p)$ for $p$ odd, up to isomorphism of fusion systems, we determine all saturated fusion systems $\mathcal{F}$ on $S$ with $O_p(\mathcal{F})=1$. For $p \ne 7$, all such fusion systems are realized by finite groups whereas for $p=7$ there are $29$ saturated fusion systems of which $27$ are exotic.
The background of this paper is the following: search of the minimal systems of generators for this class of group which still was not founded also problem of representation for this class of group, exploration of systems of generators for Sylow 2-subgroups $Syl_2{A_{2^k}}$ and $Syl_2{A_{n}}$ of alternating group, finding structure these subgroups. The aim of this paper is to research the structure of Sylow 2-subgroups and to construct a minimal generating system for such subgroups. In other words, the problem is not simply in the proof of existence of a generating set with elements for Sylow …
One part of Sylow's famous theorem in group theory states that the number of Sylow p-subgroups of a finite group is always congruent to 1 modulo p. Conversely, Marshall Hall has shown that not every positive integer $n\equiv 1\pmod{p}$ occurs as the number of Sylow p-subgroups of some finite group. While Hall's proof relies on deep knowledge of modular representation theory, we show by elementary means that no finite group has exactly 35 Sylow 17-subgroups.
The background of this paper is the following: search of the minimal systems of generators for this class of group which still was not founded also problem of representation for this class of group, exploration of systems of generators for Sylow 2-subgroups $Syl_2{A_{2^k}}$ and $Syl_2{A_{n}}$ of alternating group, finding structure of these subgroups. The authors of article "Structure of 2-sylow subgroup of symmetric and alternating group" U.~Dmitruk, V.~Suschansky didn't proof minimality of finding by them system of generators for such Sylow 2-subgroups of $A_n$ and structure of it were found…
Denote by $ν_p(G)$ the number of Sylow $p$-subgroups of $G$. It is not difficult to see that $ν_p(H)\leqν_p(G)$ for $H\leq G$, however $ν_p(H)$ does not divide $ν_p(G)$ in general. In this paper we reduce the question whether $ν_p(H)$ divides $ν_p(G)$ for every $H\leq G$ to almost simple groups. This result substantially generalizes the previous result by G. Navarro and also provides an alternative proof for the Navarro theorem.
Let $π$ be a set of primes and $\mathfrak{F}$ be a formation. In this article a properties of the class ${\rm w}^{*}_π\mathfrak{F}$ of all groups $G$, such that $π(G)\subseteq π(\mathfrak{F})$ and the normalizers of all Sylow $p$-subgroups of $G$ are $\mathfrak{F}$-subnormal in $G$ for every $p\inπ\capπ(G)$ are investigated. It is established that ${\rm w}^{*}_π\mathfrak{F}$ is a formation. Some hereditary saturated formations $\mathfrak{F}$ for which ${\rm w}^{*}_π\mathfrak{F}=\mathfrak{F}$ are founded.
Suppose that p is an odd prime and G is a finite group having no normal non-trivial p'-subgroup. We show that if a is an automorphism of G of p-power order centralizing a Sylow p-group of G, then a is inner. This answers a conjecture of Gross. An easy corollary is that if p is an odd prime and P is a Sylow p-subgroup of G, then the center of P is contained in the generalized Fitting subgroup of G. We give two proofs both requiring the classification of finite simple groups. For p=2, the result fails but Glauberman in 1968 proved that the square of a is inner. This answered a problem of Kourovk…
Given a finite group $G,$ let $e(G)$ be expected number of elements of $G$ which have to be drawn at random, with replacement, before a set of generators is found. If all the Sylow subgroups of $G$ can be generated by $d$ elements, then $e(G)\leq d+κ$ with $κ\sim 2.75239495.$ The number $κ$ is explicitly described in terms of the Riemann zeta function and is best possible. If $G$ is a permutation group of degree $n,$ then either $G=S_3$ and $e(G)=2.9$ or $e(G)\leq \lfloor n/2\rfloor+κ^*$ with $κ^* \sim 1.606695.$
This paper investigate bounds of the commutator width \cite {Mur} of a wreath product of two groups. The commutator width of direct limit of wreath product of cyclic groups are found. For given a permutational wreath product sequence of cyclic groups we investigate its commutator width and some properties of its commutator subgroup. It was proven that the commutator width of an arbitrary element of the wreath product of cyclic groups $C_{p_i}, \, p_i\in \mathbb{N} $ equals to 1. As a corollary, it is shown that the commutator width of Sylows $p$-subgroups of symmetric and alternating groups $p…
A diagonal base of a Sylow 2-subgroup $P_n(2)$ of symmetric group $S_{2^n}$ is a minimal generating set of this subgroup consisting of elements with only one non-zero coordinate in the polynomial representation. For different diagonal bases Cayley graphs of $P_n(2)$ may have different girths (i.e. minimal lengths of cycles) and thus be non-isomorphic. In presented paper all possible values of girths of Cayley graphs of $P_n(2)$ on diagonal bases are calculated. A criterion for whenever such Cayley graph has girth equal to 4 is presented. A lower bound for the number of different non-isomorphic…
We restrict irreducible characters of alternating groups of degree divisible by $p$ to their Sylow $p$-subgroups and study the number of linear constituents.
