A solvability criterion for finite groups related to character degrees

Babak Miraali; Sajjad Mahmood Robati

Displaying similar documents to “A solvability criterion for finite groups related to character degrees”

On solvability of finite groups with some $s s$ -supplemented subgroups

Jiakuan Lu, Yanyan Qiu (2015)

Czechoslovak Mathematical Journal

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A subgroup $H$ of a finite group $G$ is said to be $s s$ -supplemented in $G$ if there exists a subgroup $K$ of $G$ such that $G = H K$ and $H \cap K$ is $s$ -permutable in $K$ . In this paper, we first give an example to show that the conjecture in A. A. Heliel’s paper (2014) has negative solutions. Next, we prove that a finite group $G$ is solvable if every subgroup of odd prime order of $G$ is $s s$ -supplemented in $G$ , and that $G$ is solvable if and only if every Sylow subgroup of odd order of $G$ is $s s$ -supplemented in $G$ . These results...

On dicyclic groups as inner mapping groups of finite loops

Emma Leppälä, Markku Niemenmaa (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $G$ be a finite group with a dicyclic subgroup $H$ . We show that if there exist $H$ -connected transversals in $G$ , then $G$ is a solvable group. We apply this result to loop theory and show that if the inner mapping group $I (Q)$ of a finite loop $Q$ is dicyclic, then $Q$ is a solvable loop. We also discuss a more general solvability criterion in the case where $I (Q)$ is a certain type of a direct product.

A Non Solvable Operator Satisfying Condition $(ψ)$

Nicolas Lerner (1992)

Publications mathématiques et informatique de Rennes

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On the nontrivial solvability of systems of homogeneous linear equations over $ℤ$ in ZFC

Jan Šaroch (2020)

Commentationes Mathematicae Universitatis Carolinae

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Motivated by the paper by H. Herrlich, E. Tachtsis (2017) we investigate in ZFC the following compactness question: for which uncountable cardinals $κ$ , an arbitrary nonempty system $S$ of homogeneous $ℤ$ -linear equations is nontrivially solvable in $ℤ$ provided that each of its subsystems of cardinality less than $κ$ is nontrivially solvable in $ℤ$ ?

On computation of minimal free resolutions over solvable polynomial algebras

Huishi Li (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let $A = K [a_{1}, ..., a_{n}]$ be a (noncommutative) solvable polynomial algebra over a field $K$ in the sense of A. Kandri-Rody and V. Weispfenning [Non-commutative Gröbner bases in algebras of solvable type, J. Symbolic Comput. 9 (1990), 1–26]. This paper presents a comprehensive study on the computation of minimal free resolutions of modules over $A$ in the following two cases: (1) $A = ⨁_{p \in ℕ} A_{p}$ is an $ℕ$ -graded algebra with the degree-0 homogeneous part $A_{0} = K$ ; (2) $A$ is an $ℕ$ -filtered algebra with the filtration ${F_{p} A}_{p \in ℕ}$ determined by...

Several quantitative characterizations of some specific groups

A. Mohammadzadeh, Ali Reza Moghaddamfar (2017)

Commentationes Mathematicae Universitatis Carolinae

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Let $G$ be a finite group and let $π (G) = {p_{1}, p_{2}, ..., p_{k}}$ be the set of prime divisors of $| G |$ for which $p_{1} < p_{2} < \dots < p_{k}$ . The Gruenberg-Kegel graph of $G$ , denoted $GK (G)$ , is defined as follows: its vertex set is $π (G)$ and two different vertices $p_{i}$ and $p_{j}$ are adjacent by an edge if and only if $G$ contains an element of order $p_{i} p_{j}$ . The degree of a vertex $p_{i}$ in $GK (G)$ is denoted by $d_{G} (p_{i})$ and the $k$ -tuple $D (G) = (d_{G} (p_{1}), d_{G} (p_{2}), ..., d_{G} (p_{k}))$ is said to be the degree pattern of $G$ . Moreover, if $ω \subseteq π (G)$ is the vertex set of a connected component of $GK (G)$ , then the largest $ω$ -number which divides $| G |$ , is...

SCAP-subalgebras of Lie algebras

Sara Chehrazi, Ali Reza Salemkar (2016)

Czechoslovak Mathematical Journal

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A subalgebra $H$ of a finite dimensional Lie algebra $L$ is said to be a $SCAP$ -subalgebra if there is a chief series $0 = L_{0} \subset L_{1} \subset ... \subset L_{t} = L$ of $L$ such that for every $i = 1, 2, ..., t$ , we have $H + L_{i} = H + L_{i - 1}$ or $H \cap L_{i} = H \cap L_{i - 1}$ . This is analogous to the concept of $SCAP$ -subgroup, which has been studied by a number of authors. In this article, we investigate the connection between the structure of a Lie algebra and its $SCAP$ -subalgebras and give some sufficient conditions for a Lie algebra to be solvable or supersolvable.

The unit groups of semisimple group algebras of some non-metabelian groups of order $144$

Gaurav Mittal, Rajendra K. Sharma (2023)

Mathematica Bohemica

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We consider all the non-metabelian groups $G$ of order $144$ that have exponent either $36$ or $72$ and deduce the unit group $U (𝔽_{q} G)$ of semisimple group algebra $𝔽_{q} G$ . Here, $q$ denotes the power of a prime, i.e., $q = p^{r}$ for $p$ prime and a positive integer $r$ . Up to isomorphism, there are $6$ groups of order $144$ that have exponent either $36$ or $72$ . Additionally, we also discuss how to simply obtain the unit groups of the semisimple group algebras of those non-metabelian groups of order $144$ that are a direct product of two...

On sharp characters of type ${- 1, 0, 2}$

Alireza Abdollahi, Javad Bagherian, Mahdi Ebrahimi, Maryam Khatami, Zahra Shahbazi, Reza Sobhani (2022)

Czechoslovak Mathematical Journal

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For a complex character $χ$ of a finite group $G$ , it is known that the product $sh (χ) = \prod_{l \in L (χ)} (χ (1) - l)$ is a multiple of $| G |$ , where $L (χ)$ is the image of $χ$ on $G - {1}$ . The character $χ$ is said to be a sharp character of type $L$ if $L = L (χ)$ and $sh (χ) = | G |$ . If the principal character of $G$ is not an irreducible constituent of $χ$ , then the character $χ$ is called normalized. It is proposed as a problem by P. J. Cameron and M. Kiyota, to find finite groups $G$ with normalized sharp characters of type ${- 1, 0, 2}$ . Here we prove that such a group with nontrivial...

Thompson’s conjecture for the alternating group of degree $2 p$ and $2 p + 1$

Azam Babai, Ali Mahmoudifar (2017)

Czechoslovak Mathematical Journal

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For a finite group $G$ denote by $N (G)$ the set of conjugacy class sizes of $G$ . In 1980s, J. G. Thompson posed the following conjecture: If $L$ is a finite nonabelian simple group, $G$ is a finite group with trivial center and $N (G) = N (L)$ , then $G ≅ L$ . We prove this conjecture for an infinite class of simple groups. Let $p$ be an odd prime. We show that every finite group $G$ with the property $Z (G) = 1$ and $N (G) = N (A_{i})$ is necessarily isomorphic to $A_{i}$ , where $i \in {2 p, 2 p + 1}$ .