ABSTRACT Let G be a finite solvable group of automorphisms of a finite group K, with |G| and |K| coprime. Then there exist x,y∈K such that C G (x)∩C G (y)=1. In particular, there is an orbit of the action of G on K of size at least |G|,... more
ABSTRACT Let G be a finite solvable group of automorphisms of a finite group K, with |G| and |K| coprime. Then there exist x,y∈K such that C G (x)∩C G (y)=1. In particular, there is an orbit of the action of G on K of size at least |G|, which answers a question of I. M. Isaacs [Proc. Am. Math. Soc. 127, No. 1, 45-50 (1999; Zbl 0916.20011)]. Furthermore, in any π-solvable group, the largest normal π-subgroup is the intersection of at most three Hall π-subgroups, which generalizes a result of D. S. Passman [Trans. Am. Math. Soc. 123, 99-111 (1966; Zbl 0139.01801)]. The key to the proof is a theorem which states that if G is any solvable primitive subgroup of GL(d,p), p a prime, d a positive integer, and we denote by V the natural module for G, then either G has at least p regular orbits on V⊕V, or p≤3 and d≤6 and G is one of a small number of possible groups. The proof of this uses the computer algebra system GAP for the analysis of some small cases.
Research Interests:
In 1968, John Thompson proved that a finite group G is solvable if and only if every 2-generator subgroup of G is solvable. In this paper, we prove that solvability of a finite group G is guaranteed by a seemingly weaker condition: G is... more
In 1968, John Thompson proved that a finite group G is solvable if and only if every 2-generator subgroup of G is solvable. In this paper, we prove that solvability of a finite group G is guaranteed by a seemingly weaker condition: G is solvable if, for all conjugacy classes C and D of G consisting of elements of prime power order, there exist x in C and y in D with x and y generating a solvable group. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the solvability criterion: if G is a finite nonabelian simple group, then there exist two prime divisors a and b of |G| such that, for all elements x, y in G with |x|=a and |y|=b, the subgroup generated by x and y is not solvable. Further, using a recent result of Guralnick and Malle, we obtain a similar membership criterion for any family of finite groups closed under forming subgroups, quotients and extensions.
Research Interests:
ABSTRACT Let G be a finite group. An element g∈G is called a vanishing element of G if there exists an irreducible complex character χ of G such that χ(g)=0. The purpose of the paper under review is to continue the study, [begun in J.... more
ABSTRACT Let G be a finite group. An element g∈G is called a vanishing element of G if there exists an irreducible complex character χ of G such that χ(g)=0. The purpose of the paper under review is to continue the study, [begun in J. Group Theory 13, No. 2, 189-206 (2010; Zbl 1196.20029)], of the vanishing prime graph Γ(G), whose vertices are the prime numbers dividing the orders of some vanishing element of G, and two distinct vertices p and q are adjacent if and only if G has a vanishing element of order divisible by pq. This is a subgraph of the better-known prime graph Π(G) of a finite group which has been intensely studied over the years. The main result of the paper is that Γ(G) has at most six connected components; its proof depends on CFSG. Among other things, there is also a discussion of similarities between Γ(G) and Π(G); in particular Γ(G)=Π(G) whenever G is nonabelian simple and different from A 7 .
Research Interests:
ABSTRACT IfG is a finite group in which every element ofp′-order centralizes aq-Sylow subgroup ofG, wherep andq are distinct primes, it is shown thatO q′ (G) is solvable,l q (G)≤1 andl p (O q′ (G)) ≤2. Further, the structure ofG is... more
ABSTRACT IfG is a finite group in which every element ofp′-order centralizes aq-Sylow subgroup ofG, wherep andq are distinct primes, it is shown thatO q′ (G) is solvable,l q (G)≤1 andl p (O q′ (G)) ≤2. Further, the structure ofG is determined to some extent.
Research Interests:
We classify the C55-groups, i.e., finite groups in which the centralizer of every 5-element is a 5-group.
Research Interests:
Finally, as an application of our results, we prove that a field has at most 2 finite extensions of pairwise coprime indices with the same normal closure.
Research Interests:
Let G be a finite group and let Irr(G) denote the set of all complex irreducible characters of G. The Ito–Michler Theorem asserts that if a prime p does not divide the degree of any χ $\in$ Irr(G) then a Sylow p-subgroup P of G is normal... more
Let G be a finite group and let Irr(G) denote the set of all complex irreducible characters of G. The Ito–Michler Theorem asserts that if a prime p does not divide the degree of any χ $\in$ Irr(G) then a Sylow p-subgroup P of G is normal in G. We prove a real-valued version of this theorem, where instead of Irr(G) we only consider the subset Irrrv(G) consisting of all real-valued irreducible characters of G. We also prove that the character degree graph associated to Irrrv(G) has at most 3 connected components. Similar results for the set of real conjugacy classes of G have also been obtained.
Research Interests:
ABSTRACT Let G be a finite group. For n∈ℕ let σ(n) denote the number of distinct primes dividing n, and let σ(G)=max{σ(χ(1))∣χ∈Irr(G)} be the maximum number of primes occurring in an irreducible (complex) character degree of G. Let σ(G)... more
ABSTRACT Let G be a finite group. For n∈ℕ let σ(n) denote the number of distinct primes dividing n, and let σ(G)=max{σ(χ(1))∣χ∈Irr(G)} be the maximum number of primes occurring in an irreducible (complex) character degree of G. Let σ(G) be the set of primes dividing some irreducible character degree of G. Huppert’s ρ-σ-conjecture asserts that |ρ(G)| is bounded by some function depending only on σ(G), and more specifically, |ρ(G)|≤Cσ(G) with C=3 in general and C=2 for solvable G. So far, the solvable case has received the most attention with the currently best bound being |ρ(G)|≤3σ(G)+2 [O. Manz, T. R. Wolf, Ill. J. Math. 37, No. 4, 652-665 (1993; Zbl 0832.20005)]. In the paper under review, using the Classification of Finite Simple Groups, the authors prove that |ρ(G)|≤7σ(G) for any finite group G. They also consider the analogous (but easier) problem for conjugacy class sizes instead of character degrees, and if ρ(G) * and σ(G) * denote the corresponding quantities, then it is proved that in general |ρ(G) * |≤5σ(G) * . These linear bounds improve on the quadratic bounds for these problems obtained recently by A. Moretó [Int. Math. Res. Not. 2005, No. 54, 3375-3383 (2005; Zbl 1098.20008)]; the proofs, however, are independent of Moretó’s work.
Research Interests:
In 1968, John Thompson proved that a finite group $G$ is solvable if and only if every $2$-generator subgroup of $G$ is solvable. In this paper, we prove that solvability of a finite group $G$ is guaranteed by a seemingly weaker... more
In 1968, John Thompson proved that a finite group $G$ is solvable if and only if every $2$-generator subgroup of $G$ is solvable. In this paper, we prove that solvability of a finite group $G$ is guaranteed by a seemingly weaker condition: $G$ is solvable if for all conjugacy classes $C$ and $D$ of $G$, \emph{there exist} $x\in C$ and $y\in D$ for which $\gen{x,y}$ is solvable. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the solvability criterion: if $G$ is a finite nonabelian simple group, then there exist two integers $a$ and $b$ which represent orders of elements in $G$ and for all elements $x,y\in G$ with $|x|=a$ and $|y|=b$, the subgroup $\gen{x,y}$ is nonsolvable.
Research Interests:
Research Interests:
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ABSTRACT Suppose that G is a finite non-Abelian group with trivial Frattini subgroup and U(G) is its nilpotent residual, i.e., the smallest normal subgroup N of G such that G/N is nilpotent. It is shown that |U(G)|>2|G| |Z(G)|,... more
ABSTRACT Suppose that G is a finite non-Abelian group with trivial Frattini subgroup and U(G) is its nilpotent residual, i.e., the smallest normal subgroup N of G such that G/N is nilpotent. It is shown that |U(G)|>2|G| |Z(G)|, where Z(G) is the centre of G, and, more precisely, that |U(G)|(|U(G)|-1)≥2|G| |Z(G)| (Theorem 1). If 2∣|G| and |G| is divisible by neither a Mersenne nor a Fermat prime, then the inequality holds even without the factor 2 on its right-hand side (Theorem 2).
Research Interests:
ABSTRACT If G is finite group, a graph Γ ' (G) is defined, whose vertices are all the prime numbers which divide the length of a conjugacy class of G, and two vertices p, q are joined by an edge if there exists a conjugacy class... more
ABSTRACT If G is finite group, a graph Γ ' (G) is defined, whose vertices are all the prime numbers which divide the length of a conjugacy class of G, and two vertices p, q are joined by an edge if there exists a conjugacy class in G whose length is a multiple of pq. An old result of Ito’s says that if two primes p, q dividing the order of G are not joined in Γ ' (G), then G is either q-nilpotent or p-nilpotent. It was proved by S. Dolfi [J. Algebra 174, No. 3, 753-771 (1995; Zbl 0837.20031)] that if Γ ' (G) is not connected, then the number of connected components (which turn out to be complete) of Γ ' (G) is two, and if Γ ' (G) is connected, then its diameter is at most 3. The article characterizes all finite groups G whose associated graph Γ ' (G) is connected and has diameter 3; they are semidirect products of two abelian groups of coprime order, with some additional properties. As a by-product, some properties of connected graphs of diameter 3 which occur as graphs Γ ' (G) for some finite group G are observed; although a complete characterization of such graphs is not obtained, a comparatively large class of graphs of diameter 3 which are conjugacy class graphs Γ ' (G) for a suitable finite group G is described.Reviewer: C.Casolo (Udine)
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Let G be a finite group. An element ${g\in G}$ is a vanishing element of G if there exists an irreducible complex character χ of G such that χ(g) = 0: if this is the case, we say that the conjugacy class of g in G is a vanishing conjugacy... more
Let G be a finite group. An element ${g\in G}$ is a vanishing element of G if there exists an irreducible complex character χ of G such that χ(g) = 0: if this is the case, we say that the conjugacy class of g in G is a vanishing conjugacy class of G. In this paper we show that, if the size of every vanishing conjugacy class of G is not divisible by a given prime number p, then G has a normal p-complement and abelian Sylow p-subgroups.
Research Interests:
Research Interests:
Observe that Theorem A leaves unsettled the classification of the prime power order groups of conjugate rank 2. However, this problem is presumably quite hard. We recall that a nonabelian group is an F-group if the centralizers of its... more
Observe that Theorem A leaves unsettled the classification of the prime power order groups of conjugate rank 2. However, this problem is presumably quite hard. We recall that a nonabelian group is an F-group if the centralizers of its noncentral elements are pairwise ...
Research Interests:
Research Interests:
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... Since Out(PSU 4 (2)) is abelian, we may assume that (n,q) negationslash= (4,2), whence there exists a ppd lscript of p 2(n 1) f 1. Next, choose s GU n (q) represented ... [14]G. Navarro, Pham Huu Tiep and ... Journal of Algebra Volume... more
... Since Out(PSU 4 (2)) is abelian, we may assume that (n,q) negationslash= (4,2), whence there exists a ppd lscript of p 2(n 1) f 1. Next, choose s GU n (q) represented ... [14]G. Navarro, Pham Huu Tiep and ... Journal of Algebra Volume 323, Issue 2, 15 January 2010, Pages 540-545. ...
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We prove that if the product pq of distinct primes p and q divides the degree of some irreducible complex character of a finite group G, then pq divides the size of some conjugacy class of G.
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ABSTRACT Let G be a finite solvable group of automorphisms of a finite group K, with |G| and |K| coprime. Then there exist x,y∈K such that C G (x)∩C G (y)=1. In particular, there is an orbit of the action of G on K of size at least |G|,... more
ABSTRACT Let G be a finite solvable group of automorphisms of a finite group K, with |G| and |K| coprime. Then there exist x,y∈K such that C G (x)∩C G (y)=1. In particular, there is an orbit of the action of G on K of size at least |G|, which answers a question of I. M. Isaacs [Proc. Am. Math. Soc. 127, No. 1, 45-50 (1999; Zbl 0916.20011)]. Furthermore, in any π-solvable group, the largest normal π-subgroup is the intersection of at most three Hall π-subgroups, which generalizes a result of D. S. Passman [Trans. Am. Math. Soc. 123, 99-111 (1966; Zbl 0139.01801)]. The key to the proof is a theorem which states that if G is any solvable primitive subgroup of GL(d,p), p a prime, d a positive integer, and we denote by V the natural module for G, then either G has at least p regular orbits on V⊕V, or p≤3 and d≤6 and G is one of a small number of possible groups. The proof of this uses the computer algebra system GAP for the analysis of some small cases.
Research Interests:
We determine the structure of a finite group G whose noncentral real conjugacy classes have prime size. In particular, we show that G is solvable and that the set of the sizes of its real classes is one of the following: {1}, {1, 2}, {1,... more
We determine the structure of a finite group G whose noncentral real conjugacy classes have prime size. In particular, we show that G is solvable and that the set of the sizes of its real classes is one of the following: {1}, {1, 2}, {1, p}, or {1, 2, p}, where p is an odd prime.
Research Interests:
In 1968, John Thompson proved that a finite group G is solvable if and only if every 2-generator subgroup of G is solvable. In this paper, we prove that solvability of a finite group G is guaranteed by a seemingly weaker condition: G is... more
In 1968, John Thompson proved that a finite group G is solvable if and only if every 2-generator subgroup of G is solvable. In this paper, we prove that solvability of a finite group G is guaranteed by a seemingly weaker condition: G is solvable if, for all conjugacy classes C and D of G consisting of elements of prime power order, there exist x in C and y in D with x and y generating a solvable group. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the solvability criterion: if G is a finite nonabelian simple group, then there exist two prime divisors a and b of |G| such that, for all elements x, y in G with |x|=a and |y|=b, the subgroup generated by x and y is not solvable. Further, using a recent result of Guralnick and Malle, we obtain a similar membership criterion for any family of finite groups closed under forming subgroups, quotients and extensions.
Research Interests:
ABSTRACT Let G be a finite group. An element g∈G is called a vanishing element of G if there exists an irreducible complex character χ of G such that χ(g)=0. The purpose of the paper under review is to continue the study, [begun in J.... more
ABSTRACT Let G be a finite group. An element g∈G is called a vanishing element of G if there exists an irreducible complex character χ of G such that χ(g)=0. The purpose of the paper under review is to continue the study, [begun in J. Group Theory 13, No. 2, 189-206 (2010; Zbl 1196.20029)], of the vanishing prime graph Γ(G), whose vertices are the prime numbers dividing the orders of some vanishing element of G, and two distinct vertices p and q are adjacent if and only if G has a vanishing element of order divisible by pq. This is a subgraph of the better-known prime graph Π(G) of a finite group which has been intensely studied over the years. The main result of the paper is that Γ(G) has at most six connected components; its proof depends on CFSG. Among other things, there is also a discussion of similarities between Γ(G) and Π(G); in particular Γ(G)=Π(G) whenever G is nonabelian simple and different from A 7 .
Research Interests:
Research Interests:
ABSTRACT IfG is a finite group in which every element ofp′-order centralizes aq-Sylow subgroup ofG, wherep andq are distinct primes, it is shown thatO q′ (G) is solvable,l q (G)≤1 andl p (O q′ (G)) ≤2. Further, the structure ofG is... more
ABSTRACT IfG is a finite group in which every element ofp′-order centralizes aq-Sylow subgroup ofG, wherep andq are distinct primes, it is shown thatO q′ (G) is solvable,l q (G)≤1 andl p (O q′ (G)) ≤2. Further, the structure ofG is determined to some extent.
Research Interests:
We classify the C55-groups, i.e., finite groups in which the centralizer of every 5-element is a 5-group.
Research Interests:
Finally, as an application of our results, we prove that a field has at most 2 finite extensions of pairwise coprime indices with the same normal closure.
Research Interests:
Let G be a finite group and let Irr(G) denote the set of all complex irreducible characters of G. The Ito–Michler Theorem asserts that if a prime p does not divide the degree of any χ $\in$ Irr(G) then a Sylow p-subgroup P of G is normal... more
Let G be a finite group and let Irr(G) denote the set of all complex irreducible characters of G. The Ito–Michler Theorem asserts that if a prime p does not divide the degree of any χ $\in$ Irr(G) then a Sylow p-subgroup P of G is normal in G. We prove a real-valued version of this theorem, where instead of Irr(G) we only consider the subset Irrrv(G) consisting of all real-valued irreducible characters of G. We also prove that the character degree graph associated to Irrrv(G) has at most 3 connected components. Similar results for the set of real conjugacy classes of G have also been obtained.
Research Interests:
ABSTRACT Let G be a finite group. For n∈ℕ let σ(n) denote the number of distinct primes dividing n, and let σ(G)=max{σ(χ(1))∣χ∈Irr(G)} be the maximum number of primes occurring in an irreducible (complex) character degree of G. Let σ(G)... more
ABSTRACT Let G be a finite group. For n∈ℕ let σ(n) denote the number of distinct primes dividing n, and let σ(G)=max{σ(χ(1))∣χ∈Irr(G)} be the maximum number of primes occurring in an irreducible (complex) character degree of G. Let σ(G) be the set of primes dividing some irreducible character degree of G. Huppert’s ρ-σ-conjecture asserts that |ρ(G)| is bounded by some function depending only on σ(G), and more specifically, |ρ(G)|≤Cσ(G) with C=3 in general and C=2 for solvable G. So far, the solvable case has received the most attention with the currently best bound being |ρ(G)|≤3σ(G)+2 [O. Manz, T. R. Wolf, Ill. J. Math. 37, No. 4, 652-665 (1993; Zbl 0832.20005)]. In the paper under review, using the Classification of Finite Simple Groups, the authors prove that |ρ(G)|≤7σ(G) for any finite group G. They also consider the analogous (but easier) problem for conjugacy class sizes instead of character degrees, and if ρ(G) * and σ(G) * denote the corresponding quantities, then it is proved that in general |ρ(G) * |≤5σ(G) * . These linear bounds improve on the quadratic bounds for these problems obtained recently by A. Moretó [Int. Math. Res. Not. 2005, No. 54, 3375-3383 (2005; Zbl 1098.20008)]; the proofs, however, are independent of Moretó’s work.
Research Interests:
In 1968, John Thompson proved that a finite group $G$ is solvable if and only if every $2$-generator subgroup of $G$ is solvable. In this paper, we prove that solvability of a finite group $G$ is guaranteed by a seemingly weaker... more
In 1968, John Thompson proved that a finite group $G$ is solvable if and only if every $2$-generator subgroup of $G$ is solvable. In this paper, we prove that solvability of a finite group $G$ is guaranteed by a seemingly weaker condition: $G$ is solvable if for all conjugacy classes $C$ and $D$ of $G$, \emph{there exist} $x\in C$ and $y\in D$ for which $\gen{x,y}$ is solvable. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the solvability criterion: if $G$ is a finite nonabelian simple group, then there exist two integers $a$ and $b$ which represent orders of elements in $G$ and for all elements $x,y\in G$ with $|x|=a$ and $|y|=b$, the subgroup $\gen{x,y}$ is nonsolvable.
Research Interests:
Research Interests:
ABSTRACT Let G be a finite solvable group of automorphisms of a finite group K, with |G| and |K| coprime. Then there exist x,y∈K such that C G (x)∩C G (y)=1. In particular, there is an orbit of the action of G on K of size at least |G|,... more
ABSTRACT Let G be a finite solvable group of automorphisms of a finite group K, with |G| and |K| coprime. Then there exist x,y∈K such that C G (x)∩C G (y)=1. In particular, there is an orbit of the action of G on K of size at least |G|, which answers a question of I. M. Isaacs [Proc. Am. Math. Soc. 127, No. 1, 45-50 (1999; Zbl 0916.20011)]. Furthermore, in any π-solvable group, the largest normal π-subgroup is the intersection of at most three Hall π-subgroups, which generalizes a result of D. S. Passman [Trans. Am. Math. Soc. 123, 99-111 (1966; Zbl 0139.01801)]. The key to the proof is a theorem which states that if G is any solvable primitive subgroup of GL(d,p), p a prime, d a positive integer, and we denote by V the natural module for G, then either G has at least p regular orbits on V⊕V, or p≤3 and d≤6 and G is one of a small number of possible groups. The proof of this uses the computer algebra system GAP for the analysis of some small cases.
Research Interests:
Research Interests:
In 1968, John Thompson proved that a finite group G is solvable if and only if every 2-generator subgroup of G is solvable. In this paper, we prove that solvability of a finite group G is guaranteed by a seemingly weaker condition: G is... more
In 1968, John Thompson proved that a finite group G is solvable if and only if every 2-generator subgroup of G is solvable. In this paper, we prove that solvability of a finite group G is guaranteed by a seemingly weaker condition: G is solvable if, for all conjugacy classes C and D of G consisting of elements of prime power order, there exist x in C and y in D with x and y generating a solvable group. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the solvability criterion: if G is a finite nonabelian simple group, then there exist two prime divisors a and b of |G| such that, for all elements x, y in G with |x|=a and |y|=b, the subgroup generated by x and y is not solvable. Further, using a recent result of Guralnick and Malle, we obtain a similar membership criterion for any family of finite groups closed under forming subgroups, quotients and extensions.
Research Interests:
Research Interests:
ABSTRACT Let G be a finite group. An element g∈G is called a vanishing element of G if there exists an irreducible complex character χ of G such that χ(g)=0. The purpose of the paper under review is to continue the study, [begun in J.... more
ABSTRACT Let G be a finite group. An element g∈G is called a vanishing element of G if there exists an irreducible complex character χ of G such that χ(g)=0. The purpose of the paper under review is to continue the study, [begun in J. Group Theory 13, No. 2, 189-206 (2010; Zbl 1196.20029)], of the vanishing prime graph Γ(G), whose vertices are the prime numbers dividing the orders of some vanishing element of G, and two distinct vertices p and q are adjacent if and only if G has a vanishing element of order divisible by pq. This is a subgraph of the better-known prime graph Π(G) of a finite group which has been intensely studied over the years. The main result of the paper is that Γ(G) has at most six connected components; its proof depends on CFSG. Among other things, there is also a discussion of similarities between Γ(G) and Π(G); in particular Γ(G)=Π(G) whenever G is nonabelian simple and different from A 7 .
Research Interests:
ABSTRACT Suppose that G is a finite non-Abelian group with trivial Frattini subgroup and U(G) is its nilpotent residual, i.e., the smallest normal subgroup N of G such that G/N is nilpotent. It is shown that |U(G)|>2|G| |Z(G)|,... more
ABSTRACT Suppose that G is a finite non-Abelian group with trivial Frattini subgroup and U(G) is its nilpotent residual, i.e., the smallest normal subgroup N of G such that G/N is nilpotent. It is shown that |U(G)|>2|G| |Z(G)|, where Z(G) is the centre of G, and, more precisely, that |U(G)|(|U(G)|-1)≥2|G| |Z(G)| (Theorem 1). If 2∣|G| and |G| is divisible by neither a Mersenne nor a Fermat prime, then the inequality holds even without the factor 2 on its right-hand side (Theorem 2).
Research Interests:
ABSTRACT If G is finite group, a graph Γ ' (G) is defined, whose vertices are all the prime numbers which divide the length of a conjugacy class of G, and two vertices p, q are joined by an edge if there exists a conjugacy class... more
ABSTRACT If G is finite group, a graph Γ ' (G) is defined, whose vertices are all the prime numbers which divide the length of a conjugacy class of G, and two vertices p, q are joined by an edge if there exists a conjugacy class in G whose length is a multiple of pq. An old result of Ito’s says that if two primes p, q dividing the order of G are not joined in Γ ' (G), then G is either q-nilpotent or p-nilpotent. It was proved by S. Dolfi [J. Algebra 174, No. 3, 753-771 (1995; Zbl 0837.20031)] that if Γ ' (G) is not connected, then the number of connected components (which turn out to be complete) of Γ ' (G) is two, and if Γ ' (G) is connected, then its diameter is at most 3. The article characterizes all finite groups G whose associated graph Γ ' (G) is connected and has diameter 3; they are semidirect products of two abelian groups of coprime order, with some additional properties. As a by-product, some properties of connected graphs of diameter 3 which occur as graphs Γ ' (G) for some finite group G are observed; although a complete characterization of such graphs is not obtained, a comparatively large class of graphs of diameter 3 which are conjugacy class graphs Γ ' (G) for a suitable finite group G is described.Reviewer: C.Casolo (Udine)
Research Interests:
ABSTRACT IfG is a finite group in which every element ofp′-order centralizes aq-Sylow subgroup ofG, wherep andq are distinct primes, it is shown thatO q′ (G) is solvable,l q (G)≤1 andl p (O q′ (G)) ≤2. Further, the structure ofG is... more
ABSTRACT IfG is a finite group in which every element ofp′-order centralizes aq-Sylow subgroup ofG, wherep andq are distinct primes, it is shown thatO q′ (G) is solvable,l q (G)≤1 andl p (O q′ (G)) ≤2. Further, the structure ofG is determined to some extent.
Research Interests:
Let G be a finite group. An element ${g\in G}$ is a vanishing element of G if there exists an irreducible complex character χ of G such that χ(g) = 0: if this is the case, we say that the conjugacy class of g in G is a vanishing conjugacy... more
Let G be a finite group. An element ${g\in G}$ is a vanishing element of G if there exists an irreducible complex character χ of G such that χ(g) = 0: if this is the case, we say that the conjugacy class of g in G is a vanishing conjugacy class of G. In this paper we show that, if the size of every vanishing conjugacy class of G is not divisible by a given prime number p, then G has a normal p-complement and abelian Sylow p-subgroups.
Research Interests:
We classify the C55-groups, i.e., finite groups in which the centralizer of every 5-element is a 5-group.
Research Interests:
Research Interests:
Finally, as an application of our results, we prove that a field has at most 2 finite extensions of pairwise coprime indices with the same normal closure.
Research Interests:
Observe that Theorem A leaves unsettled the classification of the prime power order groups of conjugate rank 2. However, this problem is presumably quite hard. We recall that a nonabelian group is an F-group if the centralizers of its... more
Observe that Theorem A leaves unsettled the classification of the prime power order groups of conjugate rank 2. However, this problem is presumably quite hard. We recall that a nonabelian group is an F-group if the centralizers of its noncentral elements are pairwise ...
Research Interests:
Let G be a finite group and let Irr(G) denote the set of all complex irreducible characters of G. The Ito–Michler Theorem asserts that if a prime p does not divide the degree of any χ $\in$ Irr(G) then a Sylow p-subgroup P of G is normal... more
Let G be a finite group and let Irr(G) denote the set of all complex irreducible characters of G. The Ito–Michler Theorem asserts that if a prime p does not divide the degree of any χ $\in$ Irr(G) then a Sylow p-subgroup P of G is normal in G. We prove a real-valued version of this theorem, where instead of Irr(G) we only consider the subset Irrrv(G) consisting of all real-valued irreducible characters of G. We also prove that the character degree graph associated to Irrrv(G) has at most 3 connected components. Similar results for the set of real conjugacy classes of G have also been obtained.