On some finite 2-groups in which the derived group has two generators

Elliot Benjamin; Chip Snyder

Displaying similar documents to “On some finite 2-groups in which the derived group has two generators”

Finite-rank perturbations of positive operators and isometries

Man-Duen Choi, Pei Yuan Wu (2006)

Studia Mathematica

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We completely characterize the ranks of A - B and $A^{1 / 2} - B^{1 / 2}$ for operators A and B on a Hilbert space satisfying A ≥ B ≥ 0. Namely, let l and m be nonnegative integers or infinity. Then l = rank(A - B) and $m = r a n k (A^{1 / 2} - B^{1 / 2})$ for some operators A and B with A ≥ B ≥ 0 on a Hilbert space of dimension n (1 ≤ n ≤ ∞) if and only if l = m = 0 or 0 < l ≤ m ≤ n. In particular, this answers in the negative the question posed by C. Benhida whether for positive operators A and B the finiteness of rank(A - B) implies that...

Rank and perimeter preserver of rank-1 matrices over max algebra

Seok-Zun Song, Kyung-Tae Kang (2003)

Discussiones Mathematicae - General Algebra and Applications

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For a rank-1 matrix $A = a \otimes b^{t}$ over max algebra, we define the perimeter of A as the number of nonzero entries in both a and b. We characterize the linear operators which preserve the rank and perimeter of rank-1 matrices over max algebra. That is, a linear operator T preserves the rank and perimeter of rank-1 matrices if and only if it has the form T(A) = U ⊗ A ⊗ V, or $T (A) = U \otimes A^{t} \otimes V$ with some monomial matrices U and V.

Chern rank of complex bundle

Bikram Banerjee (2019)

Commentationes Mathematicae Universitatis Carolinae

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Motivated by the work of A. C. Naolekar and A. S. Thakur (2014) we introduce notions of upper chern rank and even cup length of a finite connected CW-complex and prove that upper chern rank is a homotopy invariant. It turns out that determination of upper chern rank of a space $X$ sometimes helps to detect whether a generator of the top cohomology group can be realized as Euler class for some real (orientable) vector bundle over $X$ or not. For a closed connected $d$ -dimensional complex manifold...

$M_{p}$ -группы с ядром произвольного ранга

В.П. Шунков (1987)

Algebra i Logika

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Random walks on finite rank solvable groups

Ch. Pittet, Laurent Saloff-Coste (2003)

Journal of the European Mathematical Society

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We establish the lower bound $p_{2 t} (e, e) ≿ exp (- t^{1 / 3})$ , for the large times asymptotic behaviours of the probabilities $p_{2 t} (e, e)$ of return to the origin at even times $2 t$ , for random walks associated with finite symmetric generating sets of solvable groups of finite Prüfer rank. (A group has finite Prüfer rank if there is an integer $r$ , such that any of its finitely generated subgroup admits a generating set of cardinality less or equal to $r$ .)

Rational realization of the minimum ranks of nonnegative sign pattern matrices

Wei Fang, Wei Gao, Yubin Gao, Fei Gong, Guangming Jing, Zhongshan Li, Yan Ling Shao, Lihua Zhang (2016)

Czechoslovak Mathematical Journal

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A sign pattern matrix (or nonnegative sign pattern matrix) is a matrix whose entries are from the set ${+, -, 0}$ ( ${+, 0}$ , respectively). The minimum rank (or rational minimum rank) of a sign pattern matrix $𝒜$ is the minimum of the ranks of the matrices (rational matrices, respectively) whose entries have signs equal to the corresponding entries of $𝒜$ . Using a correspondence between sign patterns with minimum rank $r \geq 2$ and point-hyperplane configurations in $ℝ^{r - 1}$ and Steinitz’s theorem on the rational realizability...

Class groups of large ranks in biquadratic fields

Mahesh Kumar Ram (2024)

Czechoslovak Mathematical Journal

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For any integer $n > 1$ , we provide a parametric family of biquadratic fields with class groups having $n$ -rank at least 2. Moreover, in some cases, the $n$ -rank is bigger than 4.

Infinite rank of elliptic curves over $ℚ^{a b}$

Bo-Hae Im, Michael Larsen (2013)

Acta Arithmetica

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If E is an elliptic curve defined over a quadratic field K, and the j-invariant of E is not 0 or 1728, then $E (ℚ^{a b})$ has infinite rank. If E is an elliptic curve in Legendre form, y² = x(x-1)(x-λ), where ℚ(λ) is a cubic field, then $E (K ℚ^{a b})$ has infinite rank. If λ ∈ K has a minimal polynomial P(x) of degree 4 and v² = P(u) is an elliptic curve of positive rank over ℚ, we prove that y² = x(x-1)(x-λ) has infinite rank over $K ℚ^{a b}$ .

Opérateurs sur un espace $L^{1}$

Alain Costé, Françoise Lust-Piquard (1987)

Colloquium Mathematicae

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The 4-string braid group $B_{4}$ has property RD and exponential mesoscopic rank

Sylvain Barré, Mikaël Pichot (2011)

Bulletin de la Société Mathématique de France

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We prove that the braid group $B_{4}$ on 4 strings, its central quotient $B_{4} / 〈 z 〉$ , and the automorphism group $Aut (F_{2})$ of the free group $F_{2}$ on 2 generators, have the property RD of Haagerup–Jolissaint. We also prove that the braid group $B_{4}$ is a group of intermediate mesoscopic rank (of dimension 3). More precisely, we show that the above three groups have exponential mesoscopic rank, i.e., that they contain exponentially many large flat balls which are not included in flats.

Rank α operators on the space C(T,X)

Dumitru Popa (2002)

Colloquium Mathematicae

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For 0 ≤ α < 1, an operator U ∈ L(X,Y) is called a rank α operator if $x ₙ \to^{τ_{α}} x$ implies Uxₙ → Ux in norm. We give some results on rank α operators, including an interpolation result and a characterization of rank α operators U: C(T,X) → Y in terms of their representing measures.

An alternative way to classify some Generalized Elliptic Curves and their isotopic loops

Lucien Bénéteau, M. Abou Hashish (2004)

Commentationes Mathematicae Universitatis Carolinae

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The Generalized Elliptic Curves $(GECs)$ are pairs $(Q, T)$ , where $T$ is a family of triples $(x, y, z)$ of “points” from the set $Q$ characterized by equalities of the form $x . y = z$ , where the law $x . y$ makes $Q$ into a totally symmetric quasigroup. Isotopic loops arise by setting $x * y = u . (x . y)$ . When $(x . y) . (a . b) = (x . a) . (y . b)$ , identically $(Q, T)$ is an entropic $GEC$ and $(Q, *)$ is an abelian group. Similarly, a terentropic $GEC$ may be characterized by $x^{2} . (a . b) = (x . a) (x . b)$ and $(Q, *)$ is then a Commutative Moufang Loop $(CML)$ . If in addition $x^{2} = x$ , we have Hall $GECs$ and $(Q, *)$ is an exponent $3$ $CML$ . Any...

The superfocal subgroup

Marian Deaconescu (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Nel presente lavoro vengono dimostrati teoremi d'esistenza di $p$ -complementi normali nei gruppi finiti.

On TI-subgroups and QTI-subgroups of finite groups

Ruifang Chen, Xianhe Zhao (2020)

Czechoslovak Mathematical Journal

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Let $G$ be a group. A subgroup $H$ of $G$ is called a TI-subgroup if $H \cap H^{g} = 1$ or $H$ for every $g \in G$ and $H$ is called a QTI-subgroup if $C_{G} (x) \leq N_{G} (H)$ for any $1 \neq x \in H$ . In this paper, a finite group in which every nonabelian maximal is a TI-subgroup (QTI-subgroup) is characterized.

Average r-rank Artin conjecture

Lorenzo Menici, Cihan Pehlivan (2016)

Acta Arithmetica

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Let Γ ⊂ ℚ * be a finitely generated subgroup and let p be a prime such that the reduction group Γₚ is a well defined subgroup of the multiplicative group ₚ*. We prove an asymptotic formula for the average of the number of primes p ≤ x for which [ₚ*:Γₚ] = m. The average is taken over all finitely generated subgroups $Γ = ⟨ a ₁, . . ., a_{r} ⟩ \subset ℚ *$ , with $a_{i} \in ℤ$ and $a_{i} \leq T_{i}$ , with a range of uniformity $T_{i} > e x p (4 {(l o g x l o g l o g x)}^{1 / 2})$ for every i = 1,...,r. We also prove an asymptotic formula for the mean square of the error terms in the asymptotic formula with...