scholarly journals ON VANISHING CRITERIA THAT CONTROL FINITE GROUP STRUCTURE II

2018 ◽  
Vol 98 (2) ◽  
pp. 251-257 ◽  
Author(s):  
JULIAN BROUGH ◽  
QINGJUN KONG
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Prime Power ◽  
Group Structure ◽  
Prime Power Order ◽  
Conjugacy Class Sizes ◽  
Vanishing Elements

The first author [J. Brough, ‘On vanishing criteria that control finite group structure’, J. Algebra458 (2016), 207–215] has shown that for certain arithmetical results on conjugacy class sizes it is enough to consider only the vanishing conjugacy class sizes. In this paper we further weaken the conditions to consider only vanishing elements of prime power order.

A note on class sizes of vanishing elements in finite groups

2021 ◽  
pp. 2250067
Author(s):  
Qingjun Kong ◽  
Shi Chen
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Prime Power ◽  
Prime Power Order ◽  
Normal Subgroups ◽  
Conjugacy Class Sizes ◽  
Vanishing Elements ◽  
A Finite Group

Let [Formula: see text] and [Formula: see text] be normal subgroups of a finite group [Formula: see text]. We obtain th supersolvability of a factorized group [Formula: see text], given that the conjugacy class sizes of vanishing elements of prime-power order in [Formula: see text] and [Formula: see text] are square-free.

ON CONJUGACY CLASS SIZES AND CHARACTER DEGREES OF FINITE GROUPS

2013 ◽  
Vol 13 (02) ◽  
pp. 1350100 ◽  
Author(s):  
GUOHUA QIAN ◽  
YANMING WANG
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Prime Power ◽  
P Element ◽  
Character Degrees ◽  
Prime Power Order ◽  
Conjugacy Class Sizes ◽  
A Finite Group

Let p be a fixed prime, G a finite group and P a Sylow p-subgroup of G. The main results of this paper are as follows: (1) If gcd (p-1, |G|) = 1 and p2 does not divide |xG| for any p′-element x of prime power order, then G is a solvable p-nilpotent group and a Sylow p-subgroup of G/Op(G) is elementary abelian. (2) Suppose that G is p-solvable. If pp-1 does not divide |xG| for any element x of prime power order, then the p-length of G is at most one. (3) Suppose that G is p-solvable. If pp-1 does not divide χ(1) for any χ ∈ Irr (G), then both the p-length and p′-length of G are at most 2.

The sub-class sizes of some elements being square free

2021 ◽  
Author(s):  
Xianhe Zhao ◽  
Yanyan Zhou ◽  
Ruifang Chen ◽  
Qin Huang
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Prime Power ◽  
Prime Factor ◽  
Subnormal Subgroup ◽  
Prime Power Order ◽  
Conjugacy Class Sizes ◽  
A Finite Group

Let [Formula: see text] be an element of a finite group [Formula: see text], and [Formula: see text] a prime factor of the order of [Formula: see text]. It is clear that there always exists a unique minimal subnormal subgroup containing [Formula: see text], say [Formula: see text]. We call the conjugacy class of [Formula: see text] in [Formula: see text] the sub-class of [Formula: see text] in [Formula: see text], see [G. Qian and Y. Yang, On sub-class sizes of finite groups, J. Aust. Math. Soc. (2020) 402–411]. In this paper, assume that [Formula: see text] is the product of the subgroups [Formula: see text] and [Formula: see text], we investigate the solvability, [Formula: see text]-nilpotence and supersolvability of the group [Formula: see text] under the condition that the sub-class sizes of prime power order elements in [Formula: see text] are [Formula: see text] free, [Formula: see text] free and square free, respectively, so that some known results relevant to conjugacy class sizes are generalized.

On Conjugacy Class Sizes of the p′-Elements with Prime Power Order

2009 ◽  
Vol 16 (04) ◽  
pp. 541-548 ◽  
Author(s):  
Xianhe Zhao ◽  
Xiuyun Guo
Keyword(s):  
Conjugacy Class ◽  
Solvable Group ◽  
Class Size ◽  
Prime Power ◽  
Prime Power Order ◽  
Conjugacy Class Sizes ◽  
Fixed Integer ◽  
P Elements ◽  
Conjugacy Class Size

In this paper we prove that a finite p-solvable group G is solvable if its every conjugacy class size of p′-elements with prime power order equals either 1 or m for a fixed integer m. In particular, G is 2-nilpotent if 4 does not divide every conjugacy class size of 2′-elements with prime power order.

scholarly journals ITÔ’S THEOREM ON GROUPS WITH TWO CLASS SIZES REVISITED

2011 ◽  
Vol 85 (3) ◽  
pp. 476-481 ◽  
Author(s):  
ELENA ALEMANY ◽  
ANTONIO BELTRÁN ◽  
MARÍA JOSÉ FELIPE
Keyword(s):  
Conjugacy Class ◽  
Solvable Group ◽  
Prime Power ◽  
Prime Power Order ◽  
Conjugacy Class Sizes

AbstractLet G be a finite p-solvable group. We prove that if G has exactly two conjugacy class sizes of p′-elements of prime power order, say 1 and m, then m=paqb, for two distinct primes p and q, and G either has an abelian p-complement or G=PQ×A, with P and Q a Sylow p-subgroup and a Sylow q-subgroup of G, respectively, and A is abelian. In particular, we provide a new extension of Itô’s theorem on groups having exactly two class sizes for elements of prime power order.

scholarly journals PGL2(q) cannot be determined by its cs

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1713-1719
Author(s):  
Neda Ahanjideh
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Prime Power ◽  
Simple Groups ◽  
Conjugacy Class Sizes ◽  
Direct Factor ◽  
Almost Simple Groups ◽  
Trivial Element ◽  
A Finite Group

For a finite group G, let Z(G) denote the center of G and cs*(G) be the set of non-trivial conjugacy class sizes of G. In this paper, we show that if G is a finite group such that for some odd prime power q ? 4, cs*(G) = cs*(PGL2(q)), then either G ? PGL2(q) X Z(G) or G contains a normal subgroup N and a non-trivial element t ? G such that N ? PSL2(q)X Z(G), t2 ? N and G = N. ?t?. This shows that the almost simple groups cannot be determined by their set of conjugacy class sizes (up to an abelian direct factor).

On S-Quasinormal and C-Normal Subgroups of Prime Power Order in Finite Groups

2011 ◽  
Vol 18 (04) ◽  
pp. 685-692
Author(s):  
Xuanli He ◽  
Shirong Li ◽  
Xiaochun Liu
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Maximal Subgroups ◽  
Prime Power Order ◽  
Normal Subgroups ◽  
A Finite Group

Let G be a finite group, p the smallest prime dividing the order of G, and P a Sylow p-subgroup of G with the smallest generator number d. Consider a set [Formula: see text] of maximal subgroups of P such that [Formula: see text]. It is shown that if every member [Formula: see text] of is either S-quasinormally embedded or C-normal in G, then G is p-nilpotent. As its applications, some further results are obtained.

scholarly journals Automorphism orbits of finite groups

1986 ◽  
Vol 40 (2) ◽  
pp. 253-260 ◽  
Author(s):  
Thomas J. Laffey ◽  
Desmond MacHale
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Finite Groups ◽  
Prime Power ◽  
Structure Theorem ◽  
Equivalence Classes ◽  
Prime Power Order ◽  
A Finite Group

AbstractLet G be a finite group and let Aut(G) be its automorphism group. Then G is called a k-orbit group if G has k orbits (equivalence classes) under the action of Aut(G). (For g, hG, we have g ~ h if ga = h for some Aut(G).) It is shown that if G is a k-orbit group, then kGp + 1, where p is the least prime dividing the order of G. The 3-orbit groups which are not of prime-power order are classified. It is shown that A5 is the only insoluble 4-orbit group, and a structure theorem is proved about soluble 4-orbit groups.

The Laitinen Conjecture for finite non-solvable groups

2012 ◽  
Vol 56 (1) ◽  
pp. 303-336 ◽  
Author(s):  
Krzysztof Pawałowski ◽  
Toshio Sumi
Keyword(s):  
Finite Group ◽  
Fixed Points ◽  
Solvable Group ◽  
Prime Power ◽  
Solvable Groups ◽  
Conjugacy Classes ◽  
Prime Power Order ◽  
Algebraic Condition ◽  
Smooth Action

AbstractFor any finite group G, we impose an algebraic condition, the Gnil-coset condition, and prove that any finite Oliver group G satisfying the Gnil-coset condition has a smooth action on some sphere with isolated fixed points at which the tangent G-modules are not isomorphic to each other. Moreover, we prove that, for any finite non-solvable group G not isomorphic to Aut(A6) or PΣL(2, 27), the Gnil-coset condition holds if and only if rG ≥ 2, where rG is the number of real conjugacy classes of elements of G not of prime power order. As a conclusion, the Laitinen Conjecture holds for any finite non-solvable group not isomorphic to Aut(A6).

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