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Let 휎 be a partition of the set of prime numbers. In this paper, we describe the finite groups for which every 휎-subnormal subgroup is modular. [ABSTRACT FROM AUTHOR]

Let A be a finite nilpotent group acting fixed point freely on the finite (solvable) group G by automorphisms. It is conjectured that the nilpotent length of G is bounded above by ℓ (A) , the number of primes dividing the order of A counted with multiplicities. In the present paper we consider the case A is cyclic and obtain that the nilpotent length of G is at most 2 ℓ (A) if | G | is odd. More generally we prove that the nilpotent length of G is at most 2 ℓ (A) + c (G ; A) when G is of odd order and A normalizes a Sylow system of G where c (G ; A) denotes the number of trivial A -modules appearing in an A -composition series of G. [ABSTRACT FROM AUTHOR]

We prove that there exists a universal constant D such that if p is a prime divisor of the index of the Fitting subgroup of a finite group G , then the number of conjugacy classes of G is at least $Dp/\log_2p$. We conjecture that we can take $D=1$ and prove that for solvable groups, we can take $D=1/3$. [ABSTRACT FROM AUTHOR]

Given a finite group \mathrm {G}, the faithful dimension of \mathrm {G} over \mathbb {C}, denoted by m_\mathrm {faithful}(\mathrm {G}), is the smallest integer n such that \mathrm {G} can be embedded in \mathrm {GL}_n(\mathbb {C}). Continuing the work initiated by Bardestani et al. [Compos. Math. 155 (2019), pp. 1618–1654], we address the problem of determining the faithful dimension of a finite p-group of the form \mathscr {G}_R≔\exp (\mathfrak {g}_R) associated to \mathfrak {g}_R≔\mathfrak {g}\otimes _\mathbb {Z}R in the Lazard correspondence, where \mathfrak {g} is a nilpotent \mathbb {Z}-Lie algebra and R ranges over finite truncated valuation rings. Our first main result is that if R is a finite field with p^f elements and p is sufficiently large, then m_\mathrm {faithful}(\mathscr {G}_R)=fg(p^f) where g(T) belongs to a finite list of polynomials g_1,\ldots,g_k, with non-negative integer coefficients. The latter list of polynomials is uniquely determined by the Lie algebra \mathfrak {g}. Furthermore, for each 1\le i\leq k the set of pairs (p,f) for which g=g_i is a finite union of Cartesian products \mathscr P\times \mathscr F, where \mathscr P is a Frobenius set of prime numbers and \mathscr F is a subset of \mathbb N that belongs to the Boolean algebra generated by arithmetic progressions. Previously, existence of such a polynomial-type formula for m_\mathrm {faithful}(\mathscr {G}_R) was only established under the assumption that either f=1 or p is fixed. Next we formulate a conjectural polynomiality property for the value of m_\mathrm {faithful}(\mathscr {G}_R) in the more general setting where R is a finite truncated valuation ring, and prove special cases of this conjecture. In particular, we show that for a vast class of Lie algebras \mathfrak {g} that are defined by partial orders, m_\mathrm {faithful}(\mathscr {G}_R) is given by a single polynomial-type formula. Finally, we compute m_\mathrm {faithful}(\mathscr {G}_R) precisely in the case where \mathfrak {g} is the free metabelian nilpotent Lie algebra of class c on n generators and R is a finite truncated valuation ring. [ABSTRACT FROM AUTHOR]

We discuss representations of finite groups having a common central 푝-subgroup 푍, where 푝 is a prime number. For the principal 푝-blocks, we give a method of constructing a relative 푍-stable equivalence of Morita type, which is a generalization of stable equivalence of Morita type and was introduced by Wang and Zhang in a more general setting. Then we generalize Linckelmann's results on stable equivalences of Morita type to relative 푍-stable equivalences of Morita type. We also introduce the notion of relative Brauer indecomposability, which is a generalization of the notion of Brauer indecomposability. We give an equivalent condition for Scott modules to be relatively Brauer indecomposable, which is an analog of that given by Ishioka and the first author. [ABSTRACT FROM AUTHOR]

We study the minimal number of ramified primes in Galois extensions of rational function fields over finite fields with prescribed finite Galois group. In particular, we obtain a general conjecture in analogy with the well studied case of number fields, which we establish for abelian, symmetric, and alternating groups in many cases. [ABSTRACT FROM AUTHOR]

Let ρ(n)$\rho (n)$ denote the maximal number of different primes that may occur in the order of a finite solvable group G, all elements of which have orders divisible by at most n distinct primes. We show that ρ(n)≤5n$\rho (n)\le 5n$ for all n≥1$n\ge 1$. As an application, we improve on a recent bound by Hung and Yang for arbitrary finite groups. [ABSTRACT FROM AUTHOR]

We consider G to be a finite group and p as a prime number. We fix ψ to be an irreducible character of G with its restriction to all p-regular elements of G and ψ 0 to be an irreducible Brauer character. The main aim of this paper is to describe and investigate the relationship between cyclic anchor group of ψ and the defect group of a p-block which contains ψ. Our methods are to study and generalize some facts for the cyclic defect groups of a p-block B to the case of a cyclic anchor group of irreducible characters which belong to B. We establish and prove a criteria for an irreducible character to have a cyclic anchor group. [ABSTRACT FROM AUTHOR]

It is known that, if all the real-valued irreducible characters of a finite group have odd degree, then the group has normal Sylow 2-subgroup. This result is generalized for Sylow p-subgroups, for any prime number p, while assuming the group to be p-solvable. In particular, it is proved that a p-solvable group has a normal Sylow p-subgroup if p does not divide the degree of any irreducible character of the group fixed by a field automorphism of order p. [ABSTRACT FROM AUTHOR]

Let ℓ be a prime number and let E and E ′ be ℓ -isogenous elliptic curves defined over a finite field k of characteristic p ≠ ℓ . Suppose the groups E(k) and E ′ (k) are isomorphic, but E (K) ≄ E ′ (K) , where K is an ℓ -power extension of k. In a previous work we have shown that, under mild rationality hypotheses, the case of interest is when ℓ = 2 and K is the unique quadratic extension of k. In this paper we study the likelihood of such an occurrence by fixing a pair of 2-isogenous elliptic curves E, E ′ over Q and asking for the proportion of primes p for which E (F p) ≃ E ′ (F p) and E (F p 2) ≄ E ′ (F p 2) . [ABSTRACT FROM AUTHOR]

Let G be a finite group, H a subgroup of G and p a prime number. We say that H is weakly sp-permutable in G if G has a subnormal subgroup K such that G = HK, H s G ≤ K and | H ∩ K : H s G | is a p ′ -number, where HsG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. In this paper, we investigate the structure of a group G under the assumption that certain subgroups of G are weakly sp-permutable in G. Some recent results are extended and generalized. Communicated by Alexander Olshanskii [ABSTRACT FROM AUTHOR]

Let σ = { σ i : i ∈ I } be a partition of the set of all prime numbers. A subgroup H of a finite group G is said to be σ -subnormal in G if H can be joined to G by a chain of subgroups H = H 0 ⊆ H 1 ⊆ ⋯ ⊆ H n = G where, for every j = 1 , ⋯ , n , H j − 1 is normal in H j or H j / C o r e H j (H j − 1) is a σ i -group for some i ∈ I . Let B be a subgroup of a soluble group G normalising the N σ -residual of every non- σ -subnormal subgroup of G, where N σ is the saturated formation of all σ -nilpotent groups. We show that B normalises the N σ -residual of every subgroup of G if G does not have a section that is σ -residually critical. [ABSTRACT FROM AUTHOR]

It is well known that if a finite set A ⊂ Z tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization A ⊕ B = Z M of a finite cyclic group. We are interested in characterizing all finite sets A ⊂ Z that have this property. Coven and Meyerowitz (J Algebra 212:161–174, 1999) proposed conditions (T1), (T2) that are sufficient for A to tile, and necessary when the cardinality of A has at most two distinct prime factors. They also proved that (T1) holds for all finite tiles, regardless of size. It is not known whether (T2) must hold for all tilings with no restrictions on the number of prime factors of |A|. We prove that the Coven–Meyerowitz tiling condition (T2) holds for all integer tilings of period M = (p i p j p k) 2 , where p i , p j , p k are distinct odd primes. The proof also provides a classification of all such tilings. [ABSTRACT FROM AUTHOR]

The spectra of generalized Cayley graphs of finite abelian groups are investigated in this paper. For a generalized Cayley graph X of a finite group G , the canonical double covering of X is the direct product X × K 2. In this paper, integral generalized Cayley graphs on finite abelian groups are characterized, using the characterization of the spectra of integral Cayley graphs. As an application, the integral generalized Cayley graphs on Z p × Z q and Z 2 n are investigated, where p and q are odd prime numbers. [ABSTRACT FROM AUTHOR]

Let π be a proper subset of the set of all prime numbers. Denote by r the least prime number not in π, and put m = r, if r = 2, 3, and m = r − 1 if r ≥ 5. We look at the conjecture that a conjugacy class D in a finite group G generates a π-subgroup in G (or, equivalently, is contained in the π-radical) iff any m elements from D generate a π-group. Previously, this conjecture was confirmed for finite groups whose every non-Abelian composition factor is isomorphic to a sporadic, alternating, linear or unitary simple group. Now it is confirmed for groups the list of composition factors of which is added up by exceptional groups of Lie type 2B2(q), 2G2(q), G2(q), and 3D4(q). [ABSTRACT FROM AUTHOR]

The group of rational points on an elliptic curve over Q is always a finitely generated Abelian group, hence isomorphic to Zr × G with G a finite Abelian group. Here, r is the rank of the elliptic curve. In this paper, we determine sufficient conditions that need to be set on the prime numbers p and q so that the elliptic curve E : y² = x³ − 3pqx over Q would possess a rank zero or one. Specifically, we verify that if distinct primes p and q satisfy the congruence p ≡ q ≡ 5 (mod 24), then E has rank zero. Furthermore, if p ≡ 5 (mod 12) is considered instead of a modulus of 24, then E has rank zero or one. Lastly, for primes of the form p = 24k + 17 and q = 24ℓ + 5, where 9k + 3ℓ + 7 is a perfect square, we show that E has rank one. [ABSTRACT FROM AUTHOR]

Let σ = { σ j : j ∈ J } be a partition of the set P of all prime numbers. A subgroup X of a finite group G is σ-subnormal in G if there exists a chain of subgroups X = X 0 ≤ X 1 ≤ ... ≤ X n = G such that, for each 1 ≤ i ≤ n − 1 , X i − 1 ⊴ X i or X i / (X i − 1) X i is a σ j i -group for some j i ∈ J. Skiba [18] studied the main properties of σ -subnormal subgroups in finite groups and showed that the set of all σ -subnormal subgroups plays a very relevant role in the structure of a finite soluble group. In this paper we lay the foundation of a general theory of σ -subnormal subgroups (and σ -series) in locally finite groups. Although in finite groups, σ -subnormal subgroups form a sublattice of the lattice of all subgroups (see for instance [3]), this is no longer true for locally finite groups; in fact, the join of σ -subnormal subgroups is not always σ -subnormal, but this is the case (for example) whenever the join of subnormal subgroups is subnormal (see Theorem 3.16). We provide many criteria to determining when a subgroup is σ -subnormal starting from the much weaker concept of σ-seriality (see Section 2). These criteria are particularly useful when employed to investigate the join of σ -subnormal subgroups — we show for example that if two σ -subnormal subgroups H and K of a locally finite group G are such that H K = K H , then HK is σ -subnormal in G (see Theorem 3.15) — but they are also fit to show that on some occasions σ -seriality coincides with σ -subnormality — this is the case of linear groups (see Theorem 3.35). [ABSTRACT FROM AUTHOR]

A generating set for a finite group G is minimal if no proper subset generates G, and m(G) denotes the maximal size of a minimal generating set for G. We prove a conjecture of Lucchini, Moscatiello and Spiga by showing that there exist a, b > 0 such that any finite group G satisfies m(G) 6 a · δ(G) b, for δ(G) = ∑p prime m(Gp) where Gp is a Sylow p-subgroup of G. To do this, we first bound m(G) for all almost simple groups of Lie type (until now, no nontrivial bounds were known except for groups of rank 1 or 2). In particular, we prove that there exist a, b > 0 such that any finite simple group G of Lie type of rank r over the field Fp f satisfies r + ω(f) 6 m(G) 6 a(r + ω(f))b, where ω(f) denotes the number of distinct prime divisors of f. In the process, we confirm a conjecture of Gill and Liebeck that there exist a, b > 0 such that a minimal base for a faithful primitive action of an almost simple group of Lie type of rank r over Fp f has size at most arb + ω(f). [ABSTRACT FROM AUTHOR]

Let 푝 be a prime number, let 퐻 be a finite 푝-group, and let 픽 be a field of characteristic 0, considered as a trivial F ⁢ Out ⁢ (H) -module. The main result of this paper gives the dimension of the evaluation S H , F ⁢ (G) of the simple biset functor S H , F at an arbitrary finite group 퐺. A closely related result is proved in the last section: for each prime number 푝, a Green biset functor E p is introduced, as a specific quotient of the Burnside functor, and it is shown that the evaluation E p ⁢ (G) is a free abelian group of rank equal to the number of conjugacy classes of 푝-elementary subgroups of 퐺. [ABSTRACT FROM AUTHOR]

The structure of the nonsolvable (P)-groups is completely described in this article. By definition, a finite group G is called a (P)-group if any two cyclic p-subgroups of the same order are conjugate in G, whenever p is a prime number dividing the order of G. [ABSTRACT FROM AUTHOR]

The classification of p-groups is a difficult task. It is also important because these groups have certain axioms and properties of which one can have further investigation for the other finite groups. In fact, there is no complete classification for p-groups, and the only complete classification has been done for certain cases of prime numbers or for certain nilpotency classes. In this article, we give a complete description of p-groups of order p4, where p is an odd prime. In addition, we discuss many other algebraic properties. Significantly, computer calculations were of an extensive use in this work. [ABSTRACT FROM AUTHOR]

Assume that G is a finite group. The power graph P (G) of G is a graph in which G is its node set, where two different elements are connected by an edge whenever one of them is a power of the other. A topological index is a number generated from a molecular structure that indicates important structural properties of the proposed molecule. Indeed, it is a numerical quantity connected with the chemical composition that is used to correlate chemical structures with various physical characteristics, chemical reactivity, and biological activity. This information is important for identifying well-known chemical descriptors based on distance dependence. In this paper, we study Hosoya properties, such as the Hosoya polynomial and the reciprocal status Hosoya polynomial of power graphs of various finite cyclic and non-cyclic groups of order p q and p q r , where p , q and r (p ≥ q ≥ r) are prime numbers. [ABSTRACT FROM AUTHOR]

If σ = { σ i : i ∈ I } is a partition of the set P of all prime numbers, a subgroup H of a finite group G is said to be σ -subnormal in G if H can be joined to G by means of a chain of subgroups H = H 0 ⊆ H 1 ⊆ ⋯ ⊆ H n = G such that either H i - 1 normal in H i or H i / Core H i (H i - 1) is a σ j -group for some j ∈ I , for every i = 1 , ... , n . If σ = { { 2 } , { 3 } , { 5 } ,... } is the minimal partition, then the σ -subnormality reduces to the classical subgroup embedding property of subnormality. A finite group X is said to be a Schmidt group if X is not nilpotent and every proper subgroup of X is nilpotent. Every non-nilpotent finite group G has Schmidt subgroups and a detailed knowledge of their embedding in G can provide a deep insight into its structure. In this paper, a complete description of a finite group with σ -subnormal Schmidt subgroups is given. It answers a question posed by Guo, Safonova and Skiba. [ABSTRACT FROM AUTHOR]

Let G be a finite group. For an irreducible character χ of G, the number χ c (1) = [ G : ker χ ] χ (1) is called the co-degree of χ . Let Codeg (G) denote the set of co-degrees of the irreducible (complex) characters of G. The co-degree graph of G is defined as the simple undirected graph whose vertices are prime divisors of the numbers in Codeg (G) and two distinct vertices p and q are joined by an edge if and only if pq divides some number in Codeg (G) . In this paper, we first study the finite non-solvable groups whose co-degree graphs have no complete vertices. Then, we show that if the co-degree graph of G has no complete vertices, then for every x ∈ G , G admits an irreducible character whose co-degree is divisible by o(x), the order of x. Finally, we inspect the finite groups whose co-degree graphs are m-regular. [ABSTRACT FROM AUTHOR]

Let G be a finite group. We say that an element g of G is a vanishing element if there exists an irreducible complex character of G such that (g) = 0. Denote by V o(G) the set of order of vanishing elements of G, and we prove that G ~= PSL(3, p) if and only if V o(G) = V o(PSL(3, p)) and |G| = |PSL(3, p)|, where p is a prime number. [ABSTRACT FROM AUTHOR]

Let p be a prime number. A positive integer m is said to be a p ′ -number provided p ∤ m . Let G be a finite group, and let H be a subgroup of G. We say that H is weakly s p -permutable in G if G has a subnormal subgroup K such that G = HK, H sG ≤ K and | H ∩ K : H sG | is a p ′ -number, where H sG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. We study the structure of a finite group G of even order all of whose fourth maximal subgroups are weakly s 2 -permutable in G. [ABSTRACT FROM AUTHOR]

Let G be a finite group. We say that an element g of G is a vanishing element if there exists an irreducible complex character 휒 of G such that (g) = 0. Ghasemabadi, Iranmanesh, and Mavadatpour (2015) made the following conjecture: Let G be a finite group and let M be a finite non-Abelian simple group such that Vo(G) = Vo(M) and |G| = |M|. Then G ≅ M. We give an affirmative answer to this conjecture for M = 2Dr+1(2), where r = 2n− 1 ≥ 3 and either 2r + 1 or 2r+1 + 1 is a prime number, and M = 2Dr(3), where r = 2n + 1 ≥ 5 and either (3r−1 + 1)/2 or (3r + 1)/4 is prime. [ABSTRACT FROM AUTHOR]

The spectra of generalized Cayley graphs of finite abelian groups are investigated in this paper. For a generalized Cayley graph X of a finite group G , the canonical double covering of X is the direct product X × K 2. In this paper, integral generalized Cayley graphs on finite abelian groups are characterized, using the characterization of the spectra of integral Cayley graphs. As an application, the integral generalized Cayley graphs on Z p × Z q and Z 2 n are investigated, where p and q are odd prime numbers. [ABSTRACT FROM AUTHOR]

We classify the finite groups with the property that any two different character codegrees are coprime. In general, we conjecture that if k is a positive integer such that for any prime p the number of character codegrees of a finite group G that are divisible by p is at most k, then the number of prime divisors of | G | {|G|} is bounded in terms of k. We prove this conjecture for solvable groups. [ABSTRACT FROM AUTHOR]

Let p be a prime number and let k be an algebraically closed field of characteristic p. A BT_1 group scheme over k is a finite commutative group scheme which arises as the kernel of p on a p-divisible (Barsotti–Tate) group. Our main result is that every BT_1 group scheme over k occurs as a direct factor of the p-torsion group scheme of the Jacobian of an explicit curve defined over {\mathbb F}_p. We also treat a variant with polarizations. Our main tools are the Kraft classification of BT_1 group schemes, a theorem of Oda, and a combinatorial description of the de Rham cohomology of Fermat curves. [ABSTRACT FROM AUTHOR]

In this paper, we prove that projective special linear groups PSL(5, 2) and PSL(4, 5) can be uniquely determined by their order and the largest order of elements. [ABSTRACT FROM AUTHOR]

A result of Gluck is that any finite group G has an abelian subgroup A such that |G : A| is bounded by a polynomial function of the largest irreducible character degree of G. Moretó presented a variation of this result that looks at the number of prime factors of the irreducible character degrees and obtained an almost quadratic bound. The author improved the result of Moretó to almost linear. In this note, we further improve the bound, and also study the related problem on conjugacy class sizes. [ABSTRACT FROM AUTHOR]

Given a finite group G , the character graph , denoted by Δ (G) , for its irreducible character degrees is a graph with vertex set ρ (G) which is the set of prime numbers that divide the irreducible character degrees of G , and with { p , q } being an edge if there exists a non-linear χ ∈ Irr (G) whose degree is divisible by p q. In this paper, on one hand, we proceed by discussing the graphical shape of Δ (G) when it has cut vertices or small number of eigenvalues, and on the other hand we give some results on the group structure of G with such Δ (G). Recently, Lewis and Meng proved the character graph of each solvable group has at most one cut vertex. Now, we determine the structure of character graphs of solvable groups with a cut vertex and diameter 3. Furthermore, we study solvable groups whose character graphs have at most two distinct eigenvalues. Moreover, we investigate the solvable groups whose character graphs are regular with three distinct eigenvalues. In addition, we give some lower bounds for the number of edges of Δ (G). [ABSTRACT FROM AUTHOR]

The order prime divisor graph 𝒫𝒟(G) of a finite group G is a simple graph whose vertex set is G and two vertices a, b ∈ G are adjacent if and only if either ab = e or o(ab) is some prime number, where e is the identity element of the group G and o(x) denotes the order of an element x ∈ G. In this paper, we establish the necessary and sufficient condition for the completeness of order prime divisor graph 𝒫𝒟(G) of a group G. Concentrating on the graph 𝒫𝒟(Dn), we investigate several properties like degrees, girth, regularity, Eulerianity, Hamiltonicity, planarity etc. We characterize some graph theoretic properties of 𝒫𝒟 (ℤn), 𝒫𝒟 (Sn), 𝒫𝒟 (An). [ABSTRACT FROM AUTHOR]

The number of different prime divisors of the order of a finite group G is bounded above by a quartic function of the maximum number of different prime divisors of the order of a single element in G. This improves earlier results of J. Zhang, T. M. Keller, and A. Moretó. [ABSTRACT FROM AUTHOR]

Let G be the simple group PSL (3 , 2 p) , where p is a prime number. For any subgroup H of G , we compute the Möbius function μ (H) of H in the subgroup lattice of G. To this aim, we describe the intersections of maximal subgroups of G. We point out some connections of the Möbius function with other combinatorial objects, and, in this context, we compute the reduced Euler characteristic of the order complex of the subposet of r -subgroups of PGL (3 , q) , for any prime r and any prime power q. [ABSTRACT FROM AUTHOR]

Let k be a number field, O k its ring of integers, Cl (k) its classgroup and h the class number of k. Let Γ be a finite group. Let ℳ be a maximal O k -order in the semi-simple algebra k [ Γ ] containing O k [ Γ ] , and Cl (ℳ) its locally free classgroup. Let A = ℤ ∪ { 1 m , m ∈ ℤ ∖ { 0 , 1 , − 1 } } and n ∈ A. We define the set ℛ ( n , ℳ) of Galois module classes realizable by the n th power of the different to be the set of classes c ∈ Cl (ℳ) such that there exists a Galois extension N / k with Galois group isomorphic to Γ (Γ -extension), which is tamely ramified, and for which the class of ℳ ⊗ O k [ Γ ] N / k n is equal to c , where we clarify that if n = 1 m , where m ∈ ℤ ∖ { 0 , 1 , − 1 } , N / k n is the | m | th root of the inverse different N / k − 1 (respectively, the different N / k ) if m < 0 (respectively, m > 0) when it exists. Let l be a prime number and ξ be a primitive l th root of unity. In this article, we suppose that Γ is cyclic of order l and k / ℚ and ℚ (ξ) / ℚ are linearly disjoint. We prove, sometimes under an assumption on h , that ℛ ( n , ℳ) is a subgroup of Cl (ℳ) , by an explicit description using a Stickelberger ideal. In addition, for each n ∈ A , we determine the set of the Steinitz classes of N / k n , N / k runs through the tame Γ -extensions of k , and prove that it is a subgroup of Cl (k) , also sometimes under an hypothesis on h. [ABSTRACT FROM AUTHOR]

Elementary abelian groups are finite groups in the form of A = (ℤ/pℤ) r for a prime number p. For every integer ℓ > 1 and r > 1, we prove a non-trivial upper bound on the ℓ-torsion in class groups of every A-extension. Our results are pointwise and unconditional. This establishes the first case where for some Galois group G, the ℓ-torsion in class groups are bounded non-trivially for every G-extension and every integer ℓ > 1. When r is large enough, the unconditional pointwise bound we obtain also breaks the previously best known bound shown by Ellenberg and Venkatesh under GRH. [ABSTRACT FROM AUTHOR]

Let ℙ be the set of all prime numbers. A subgroup H of a finite group G is called ℙ-subnormal if either H = G or there exists a chain of subgroups H = H0 ≤ H1 ≤ ... ≤ Hn = G such that |Hi : Hi − 1| ∈ ℙ, 1 ≤ i ≤ n. We prove that any finite group with ℙ-subnormal Sylow p-subgroup of odd order is p-solvable and any group with ℙ-subnormal generalized Schmidt subgroups is metanilpotent. [ABSTRACT FROM AUTHOR]

Let n be a positive integer and p a prime number. The number of finite abelian groups of order pn is given by the partition function p(n). The number of subgroups of a fixed order in a finite abelian group of given rank is given by sums of Hall polynomials. Here, we use recurrence relations to derive explicit formulas for counting the number of subgroups of given order (or index) in rank 3 finite abelian p-groups and use these to derive similar formulas in few cases for rank 4. As a consequence, we answer some questions by M. Tärnäuceanu and L. Tóth. We also use other methods such as the method of fundamental group lattices introduced by Tärnäuceanu to derive a similar counting function in a special case of arbitrary rank finite abelian p-groups. [ABSTRACT FROM AUTHOR]

Let p be a prime number and let k be an algebraically closed field of characteristic p. A BT1 group scheme over k is a finite commutative group scheme which arises as the kernel of p on a p-divisible (Barsotti–Tate) group. We compare three classifications of BT1 group schemes, due in large part to Kraft, Ekedahl, and Oort, and defined using words, canonical filtrations, and permutations. Using this comparison, we determine the Ekedahl–Oort types of Fermat quotient curves and we compute four invariants of the p-torsion group schemes of these curves. [ABSTRACT FROM AUTHOR]

Let q be an odd prime power and n an integer. Let ℓ ∈ F q n [ x ] be a q -linearized t -scattered polynomial of linearized degree r. Let d = max ⁡ { t , r } be an odd prime number. In this paper we show that under these assumptions it follows that ℓ = x. Our technique involves a Galois theoretical characterization of t -scattered polynomials combined with the classification of transitive subgroups of the general linear group over the finite field F q. [ABSTRACT FROM AUTHOR]

A finite group G is called (l,m, n)-generated, if it is a quotient group of the triangle group T (l,m, n) = . In [24], Moori posed the question of finding all the (p, q, r) triples, where p, q and r are prime numbers, such that a non-abelian finite simple group G is a (p, q, r)-generated. In this paper we will establish all the (p, q, r)-generations of the Mathieu group M22. GAP [16] and the Atlas of finite group representations [30] are used in our computations. [ABSTRACT FROM AUTHOR]

Let G be a finite group, and p a prime number; a character of G is called p-constant if it takes a constant value on all the elements of G whose order is divisible by p. This is a generalization of the very important concept of characters of p-defect zero. In this paper, we characterize the finite p-solvable groups having a faithful irreducible character that is p-constant and not of p-defect zero, and we will show that a non- p-solvable group with this property is an almost-simple group. [ABSTRACT FROM AUTHOR]

Let G and A be finite groups with A acting on G by automorphisms. In this paper we introduce the concept of "good action"; namely we say the action of A on G is good, if H = H , B C H (B) for every subgroup B of A and every B -invariant subgroup H of G. This definition allows us to prove a new noncoprime Hall-Higman type theorem. If A is a nilpotent group acting on the finite solvable group G with C G (A) = 1 , a long standing conjecture states that h (G) ⩽ ℓ (A) where h (G) is the Fitting height of G and ℓ (A) is the number of primes dividing the order of A counted with multiplicities. As an application of our result we prove the main theorem of this paper which states that the above conjecture is true if A and G have odd order, the action of A on G is good and some other fairly general conditions are satisfied. [ABSTRACT FROM AUTHOR]

Let σ = { σ i : i ∈ I } be a partition of the set P of all prime numbers. A subgroup X of a finite group G is called σ - subnormal in G if there is chain of subgroups X = X 0 ⊆ X 1 ⊆ ⋯ ⊆ X n = G with X i − 1 normal in X i or X i / C o r e X i (X i − 1) is a σ i -group for some i ∈ I , 1 ≤ i ≤ n. In the special case that σ is the partition of P into sets containing exactly one prime each, the σ -subnormality reduces to the familiar case of subnormality. If a finite soluble group G = A B is factorised as the product of the subgroups A and B , and X is a subgroup of G such that X is σ -subnormal in 〈 X , X g 〉 for all g ∈ A ∪ B , we prove that X is σ -subnormal in G. This is an extension of a subnormality criteria due to Maier and Sidki and Casolo. [ABSTRACT FROM AUTHOR]

In this paper, we determine all of finite groups whose order and the largest of their irreducible character degrees are the same as PGL (2 , p 2) for all odd prime numbers p. As a consequence, we show that the groups PGL (2 , p 2) are uniquely determined by the structure of their complex group algebras. [ABSTRACT FROM AUTHOR]