Search results for "Prime number"
showing 10 items of 26 documents
Permutability of injectors with a central socle in a finite solvable group
2017
In response to an Open Question of Doerk and Hawkes [5, IX Section 3, page 615], we shall show that if Zπ is the Fitting class formed by the finite solvable groups whose π-socle is central (where π is a set of prime numbers), then the Zπ-injectors of a finite solvable group G permute with the members of a Sylow basis in G. The proof depends on the properties of certain extraspecial groups [4].
Injectors with a central socle in a finite solvable group
2013
Abstract In response to an Open Question of Doerk and Hawkes (1992) [2, IX §4, p. 628] , we shall describe three constructions for the Z π -injectors of a finite solvable group, where Z π is the Fitting class formed by the finite solvable groups whose π -socle is central (and π is a set of prime numbers).
A simple proof of the polylog counting ability of first-order logic
2007
The counting ability of weak formalisms (e.g., determining the number of 1's in a string of length N ) is of interest as a measure of their expressive power, and also resorts to complexity-theoretic motivations: the more we can count the closer we get to real computing power. The question was investigated in several papers in complexity theory and in weak arithmetic around 1985. In each case, the considered formalism (AC 0 -circuits, first-order logic, Δ 0 ) was shown to be able to count up to a polylogarithmic number. An essential part of the proofs is the construction of a 1-1 mapping from a small subset of {0, ..., N - 1} into a small initial segment. In each case the expressibility of …
On σ-subnormal closure
2020
Let σ={σi:i∈I} be a partition of the set P of all prime numbers. A subgroup A of a finite group G is called σ-subnormal in G if there is a chain of subgroups A=A0⊆A1⊆⋯⊆An=G with Ai−1 normal in Ai o...
On σ-subnormality criteria in finite groups
2022
Abstract Let σ = { σ i : i ∈ I } be a partition of the set P of all prime numbers. A subgroup H of a finite group G is called σ-subnormal in G if there is a chain of subgroups H = H 0 ⊆ H 1 ⊆ ⋯ ⊆ H n = G where, for every i = 1 , … , n , H i − 1 normal in H i or H i / C o r e H i ( H i − 1 ) is a σ j -group for some j ∈ I . 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. In this paper some σ-subnormality criteria for subgroups of finite groups are studied.
Number of Sylow subgroups in $p$-solvable groups
2003
If G is a finite group and p is a prime number, let vp(G) be the number of Sylow p-subgroups of G. If H is a subgroup of a p-solvable group G, we prove that v p (H) divides v p (G).
A submatrix of the character table
2000
Let G be a finite group and let p be a prime number. We consider the Submatrix of the character table of G whose rows are indexed by the characters in blocks of maximal defect, and whose columns are indexed by the conjugacy classes of P′-size. We prove that this matrix has maximum rank.
A Question of R. Maier Concerning Formations
1996
The formation f is said to be saturated if the group G belongs to f Ž . whenever the Frattini factor group GrF G is in f. Let P be the set of all prime numbers. A formation function is a Ž . function f defined on P such that f p is a, possibly empty, formation. A formation f is said to be a local formation if there exists a formation Ž function f such that f s G g G : if HrK is a chief factor of G and p < < Ž . Ž .. divides HrK , then GrC HrK g f p ; G is the class of all finite G groups. If f is a local formation defined by a formation function f , then Ž . we denote f s LF f and f is a local definition of f. Among all possible local definitions of a local formation f there exists exactly …
Capabilities of Ultrametric Automata with One, Two, and Three States
2016
Ultrametric automata use p-adic numbers to describe the random branching of the process of computation. Previous research has shown that ultrametric automata can have a significant decrease in computing complexity. In this paper we consider the languages that can be recognized by one-way ultrametric automata with one, two, and three states. We also show an example of a promise problem that can be solved by ultrametric integral automaton with three states.
A reduction theorem for a conjecture on products of two π -decomposable groups
2013
[EN] For a set of primes pi, a group X is said to be pi-decomposable if X = X-pi x X-pi' is the direct product of a pi-subgroup X-pi and a pi'-subgroup X-pi', where pi' is the complementary of pi in the set of all prime numbers. The main result of this paper is a reduction theorem for the following conjecture: "Let pi be a set of odd primes. If the finite group G = AB is a product of two pi-decomposable subgroups A = A(pi) x A(pi') and B = B-pi x B-pi', then A(pi)B(pi) = B(pi)A(pi) and this is a Hall pi-subgroup of G." We establish that a minimal counterexample to this conjecture is an almost simple group. The conjecture is then achieved in a forthcoming paper. (C) 2013 Elsevier Inc. All ri…