Sylow Subgroups Research Articles

Normalizers of Sylow subgroups in finite reflection groups

Let W be a finite reflection group, either real or complex, and S l a Sylow l -subgroup of W. We prove the existence of a semidirect product decomposition of N W ( S l ) in terms of the unique parabolic subgroup of W minimally containing S l and known decompositions of normalizers of parabolic subgroups. In the real setting, the description follows from the existence of Sylow l -subgroups stable under the Coxeter diagram automorphisms of finite reflection groups with no proper parabolic subgroup containing a Sylow l -subgroup.

On semiabelian groups

Abstract Semiabelian groups are a family of finite solvable groups that have played an important role in the inverse Galois problem and the minimal ramification problem. In this paper, we establish several sufficient conditions for a solvable group to be semiabelian mainly by using its Sylow subgroups. We also present a family of non-semiabelian 2-groups.

Homotopy classification of 4-manifolds with finite abelian 2-generator fundamental groups

Abstract We show that for an oriented 4-dimensional Poincaré complex X with finite fundamental group, whose 2-Sylow subgroup is abelian with at most 2 generators, the homotopy type of X is determined by its quadratic 2-type.

Blocks whose defect groups are Suzuki 2-groups

We classify up to Morita equivalence all blocks whose defect groups are Suzuki 2-groups. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field, and in fact holds up to basic Morita equivalence. As a consequence Donovan's conjecture holds for Suzuki 2-groups. A corollary of the proof is that Suzuki Sylow 2-subgroups of finite groups with no nontrivial odd order normal subgroup are trivial intersection.

Fixed point subgroups of a supertight automorphism

Let G be an infinite simple group of finite Morley rank and α a supertight automorphism of G so that the fixed point subgroup P n : = C G ( α n ) is pseudofinite for all n ∈ N ∖ { 0 } . It is know (using CFSG) that the socle S n : = Soc ( P n ) is a (twisted) Chevalley group over a pseudofinite field. We prove that there is r ∈ N ∖ { 0 } so that for each n we have [ P n : S n ] < r and that there is no m ∈ N ∖ { 0 } so that for each n the sizes of the Sylow 2-subgroups of S n are bounded by m. We also note that in the recent identification result of G under the assumption pr 2 ( G ) = 1 , the use of CFSG is not needed.

Infinite locally finite groups groups with the given properties of the norm of Abelian non-cyclic subgroups

In the paper, the properties of infinite locally finite groups with non-Dedekind locally nil\-potent norms of Abelian non-cyclic subgroups are studied. It is proved that such groups are finite extensions of a quasicyclic subgroup and contain Abelian non-cyclic $p$-subgroups for a unique prime $p$. In particular, in the paper is prove the following assertions: 1) Let $G$ be an infinite locally finite group and contain the locally nilpotent norm $N_{G}^{A}$ with the non-Hamiltonian Sylow $p$-subgroup $(N_{G}^{A})_{p}$. Then $G$ is a finite extension of a quasicyclic $p$-subgroup, all Sylow $p'$-subgroups are finite and do not contain Abelian non-cyclic subgroups. In particular, Sylow $q$-subgroups ($q$ is an odd prime, $q\in \pi(G)$, $q\neq p$) are cyclic, Sylow $2$-subgroups ($p\neq 2$) are either cyclic or finite quaternion $2$-groups (Theorem 1). 2) Let $G$ be a locally finite non-locally nilpotent group with the infinite locally nilpotent non-Dedekind norm $N_{G}^{A}$ of Abelian non-cyclic subgroups. Then $G=G_{p} \leftthreetimes H,$ where $G_{p}$ is an infinite $\overline{HA}_{p}$-group of one of the types (1)--(4) of Proposition~2 in present paper, which coincides with the Sylow $p$-subgroup of the norm $N_{G}^{A}$, $H$ is a finite group, all Abelian subgroups of which are cyclic, and $(|H|,p)=1$. Any element $h\in H$ of odd order that centralizes some Abelian non-cyclic subgroup $M\subset N_{G}^{A}$ is contained in the centralizer of the norm $N_{G}^{A}$. (Theorem 2).3) Let $G$ be an infinite locally finite non-locally nilpotent group with the finite nilpotent non-Dedekind norm $N_{G}^{A}$ of Abelian non-cyclic subgroups. Then$G=H\leftthreetimes K,$ where $H$ is a finite group, all Abelian subgroups of which are cyclic,$\left(\left|H\right|,2\right)=1$, $K$ is an infinite 2-group of one of the types (5)--(6) of Proposition~2 (in present paper). Moreover, the norm $N_{K}^{A}$ of Abelian non-cyclic subgroups of the group $K$ is finite, $K\cap N_{G}^{A}=N_{K}^{A}$ and coincides with the Sylow 2-subgroup $(N_{G}^{A})_2$ of the norm $N_{G}^{A}$ of a group $G$.Moreover, any element $h\in H$ of the centralizer of some Abelian non-cyclic subgroup $M \subset N_{G}^{A}$ is contained in the centralizer of the norm $N_{G}^{A}$. (Theorem 4).

Fuchs' problem for endomorphisms of nonabelian groups

In 1960, László Fuchs posed the problem of determining which groups G are realizable as the group of units in some ring R. In [4], we investigated the following variant of Fuchs' problem, for abelian groups: which groups G are realized by a ring R where every group endomorphism of G is induced by a ring endomorphism of R? Such groups are called fully realizable. In this paper, we answer the aforementioned question for several families of nonabelian groups: symmetric, dihedral, quaternion, alternating, and simple groups; almost cyclic p-groups; and groups whose Sylow 2-subgroup is either cyclic or normal and abelian. We construct three infinite families of fully realizable nonabelian groups using iterated semidirect products.

On Oliver's p-group conjecture for Sylow subgroups of symplectic groups and orthogonal groups

In this paper, we focus on Oliver's p-group conjecture. We prove that Oliver's p-group conjecture holds for Sylow p-subgroups of symplectic groups and orthogonal groups.

The automorphism group of finite 2-groups associated to the Macdonald group

We consider the Macdonald group [Formula: see text] and its Sylow 2-subgroup [Formula: see text], where [Formula: see text] and [Formula: see text] is odd. Then [Formula: see text] has order [Formula: see text], and nilpotency class 5 if [Formula: see text] and 3 if [Formula: see text]. We determine the automorphism group of the 2-groups [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text]. Explicit multiplication, power, and commutator formulas for [Formula: see text], [Formula: see text] and [Formula: see text] are given, and used in the calculation of [Formula: see text], [Formula: see text] and [Formula: see text].

New spence difference sets

AbstractSpence [9] constructed $$\left( \frac{3^{d+1}(3^{d+1}-1)}{2}, \frac{3^d(3^{d+1}+1)}{2}, \frac{3^d(3^d+1)}{2}\right) $$ 3 d + 1 ( 3 d + 1 - 1 ) 2 , 3 d ( 3 d + 1 + 1 ) 2 , 3 d ( 3 d + 1 ) 2 -difference sets in groups $$K \times C_3^{d+1}$$ K × C 3 d + 1 for d any positive integer and K any group of order $$\frac{3^{d+1}-1}{2}$$ 3 d + 1 - 1 2 . Smith and Webster [8] have exhaustively studied the $$d=1$$ d = 1 case without requiring that the group have the form listed above and found many constructions. Among these, one intriguing example constructs Spence difference sets in $$A_4 \times C_3$$ A 4 × C 3 by using (3, 3, 3, 1)-relative difference sets in a non-normal subgroup isomorphic to $$C_3^2$$ C 3 2 . Drisko [3] has a note implying that his techniques allow constructions of Spence difference sets in groups with a noncentral normal subgroup isomorphic to $$C_3^{d+1}$$ C 3 d + 1 as long as $$\frac{3^{d+1}-1}{2}$$ 3 d + 1 - 1 2 is a prime power. We generalize this result by constructing Spence difference sets in similar families of groups, but we drop the requirement that $$\frac{3^{d+1}-1}{2}$$ 3 d + 1 - 1 2 is a prime power. We conjecture that any group of order $$\frac{3^{d+1}(3^{d+1}-1)}{2}$$ 3 d + 1 ( 3 d + 1 - 1 ) 2 with a normal subgroup isomorphic to $$C_3^{d+1}$$ C 3 d + 1 will have a Spence difference set (this is analogous to Dillon’s conjecture in 2-groups, and that result was proved in Drisko’s work). Finally, we present the first known example of a Spence difference set in a group where the Sylow 3-subgroup is nonabelian and has exponent bigger than 3. This new construction, found by computing the full automorphism group $$\textrm{Aut}(\mathcal {D})$$ Aut ( D ) of a symmetric design associated to a known Spence difference set and identifying a regular subgroup of $$\textrm{Aut}(\mathcal {D})$$ Aut ( D ) , uses (3, 3, 3, 1)-relative difference sets to describe the difference set.

New characteristic subgroups for p-stable finite groups and fusion systems

For a p-group S, we define a new characteristic subgroup Q(e)(S) of S. We prove that Q(e)(S) is also a characteristic subgroup for p-stable finite groups of characteristic p having S as a Sylow subgroup. The present paper also establishes that Q(e)(S) is a normal subgroup of any p-stable fusion system over S.

Finite Groups with $ G $-Permutable Normalizers of Sylow Subgroups

Finite Groups with $ G $-Permutable Normalizers of Sylow Subgroups

On the partial Π-property of second minimal or second maximal subgroups of Sylow subgroups of finite groups

Let H be a subgroup of a finite group G. We say that H satisfies the partial Π -property in G if there exists a chief series Γ G : 1 = G 0 < G 1 < · · · < G n = G of G such that for every G-chief factor G i / G i − 1 ( 1 ≤ i ≤ n ) of Γ G , | G / G i − 1 : N G / G i − 1 ( H G i − 1 / G i − 1 ∩ G i / G i − 1 ) | is a π ( H G i − 1 / G i − 1 ∩ G i / G i − 1 ) -number. In this paper, we study the influence of some second minimal or second maximal subgroups of a Sylow subgroup satisfying the partial Π -property on the structure of a finite group.

The number of conjugacy classes of noncyclic subgroups of finite nilpotent groups

Let [Formula: see text] be a finite nilpotent group. We denote by [Formula: see text] the number of conjugacy classes of noncyclic subgroups of [Formula: see text], [Formula: see text] the number of conjugacy classes of subgroups of [Formula: see text]. In this paper, we give the formula for [Formula: see text], and prove that [Formula: see text] when the order of [Formula: see text] is prime power. Moreover, we get the minimum of [Formula: see text] according to [Formula: see text], where [Formula: see text] is the number of distinct noncyclic Sylow subgroups of [Formula: see text].

Overgroup coset complexes in finite groups

Let G be a finite group and X be a subgroup of G. We investigate the topological properties of the poset CX(G) of cosets Hx in G with X≤H<G. We show that CX(G) is non-contractible if G is solvable or NG(X) contains a Sylow 2-subgroup and a Sylow 3-subgroup of G. This result follows J. Shareshian and R. Woodroofe's work in [8] (2016). We also give some divisibility properties of the Euler characteristic of CX(G) when X is a p-group, which follows K. S. Brown's classical result in [3] (2000).

On redundant Sylow subgroups

A Sylow p-subgroup P of a finite group G is called redundant if every p-element of G lies in a Sylow subgroup different from P. Generalizing a recent theorem of Maróti–Martínez–Moretó, we show that for every non-cyclic p-group P there exists a solvable group G such that P is redundant in G. Moreover, we answer several open questions raised by Maróti–Martínez–Moretó.

A CHARACTERISATION OF SOLUBLE -GROUPS

Abstract Let G be a finite group. A subgroup A of G is said to be S-permutable in G if A permutes with every Sylow subgroup P of G, that is, $AP=PA$ . Let $A_{sG}$ be the subgroup of A generated by all S-permutable subgroups of G contained in A and $A^{sG}$ be the intersection of all S-permutable subgroups of G containing A. We prove that if G is a soluble group, then S-permutability is a transitive relation in G if and only if the nilpotent residual $G^{\mathfrak {N}}$ of G avoids the pair $(A^{s G}, A_{sG})$ , that is, $G^{\mathfrak {N}}\cap A^{sG}= G^{\mathfrak {N}}\cap A_{sG}$ for every subnormal subgroup A of G.

Coleman automorphisms of finite groups with semidihedral Sylow 2-subgroups

We study some families of finite groups having inner class-preservingautomorphisms. In particular, let G be a finite group and S be asemidihedral Sylow 2-subgroup. Then, in both cases when either Sym(4) is nota homomorphic image of G and Z(S)<Z(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Z(S) < Z(G)$$\\end{document} or G is nilpotent-by-nilpotent, wehave that all the Coleman automorphisms of G are inner. As a consequence, thesegroups satisfy the normalizer problem.

The commuting graph of a solvable 𝐴-group

Abstract Let 𝐺 be a finite group. Recall that an 𝐴-group is a group whose Sylow subgroups are all abelian. In this paper, we investigate the upper bound on the diameter of the commuting graph of a solvable 𝐴-group. Assuming that the commuting graph is connected, we show when the derived length of 𝐺 is 2, the diameter of the commuting graph will be at most 4. In the general case, we show that the diameter of the commuting graph will be at most 6. In both cases, examples are provided to show that the upper bound of the commuting graph cannot be improved.

On recognition of the direct squares of the simple groups with abelian Sylow 2-subgroups

On recognition of the direct squares of the simple groups with abelian Sylow 2-subgroups