scholarly journals THE EMBEDDING OF A CYCLIC PERMUTABLE SUBGROUP IN A FINITE GROUP. II

2004 ◽  
Vol 47 (1) ◽  
pp. 101-109 ◽  
Author(s):  
J. Cossey ◽  
S. E. Stonehewer
Keyword(s):  
Finite Group ◽  
Subject Classification ◽  
Normal Closure ◽  
General Situation ◽  
Permutable Subgroup ◽  
Odd Order ◽  
Even Order ◽  
Group Ii ◽  
A Finite Group

AbstractIn two previous papers we established the structure of the normal closure of a cyclic permutable subgroup $A$ of a finite group, first when $A$ has odd order and second when $A$ has even order, but with an extra hypothesis that was unnecessary in the odd case. Here we describe the most general situation without any restrictions on $A$.AMS 2000 Mathematics subject classification: Primary 20D35; 20D40

scholarly journals Cyclic permutable subgroups of finite groups

2001 ◽  
Vol 71 (2) ◽  
pp. 169-176 ◽  
Author(s):  
John Cossey ◽  
Stewart E. Stonehewer
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Normal Closure ◽  
Permutable Subgroup ◽  
Odd Order ◽  
Permutable Subgroups ◽  
A Finite Group

AbstractThe authors describe the structure of the normal closure of a cyclic permutable subgroup of odd order in a finite group.

scholarly journals Finite Groups with All Maximal Subgroups of Prime or Prime Square Index

1964 ◽  
Vol 16 ◽  
pp. 435-442 ◽  
Author(s):  
Joseph Kohler
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Prime Order ◽  
Maximal Subgroups ◽  
Odd Order ◽  
Order Case ◽  
Chief Factors ◽  
Even Order ◽  
A Finite Group

In this paper finite groups with the property M, that every maximal subgroup has prime or prime square index, are investigated. A short but ingenious argument was given by P. Hall which showed that such groups are solvable.B. Huppert showed that a finite group with the property M, that every maximal subgroup has prime index, is supersolvable, i.e. the chief factors are of prime order. We prove here, as a corollary of a more precise result, that if G has property M and is of odd order, then the chief factors of G are of prime or prime square order. The even-order case is different. For every odd prime p and positive integer m we shall construct a group of order 2apb with property M which has a chief factor of order larger than m.

scholarly journals On characters in the principal 2-block, II

1979 ◽  
Vol 28 (1) ◽  
pp. 100-106 ◽  
Author(s):  
Marcel Herzog ◽  
Cheryl E. Praeger
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Complex Number ◽  
Irreducible Character ◽  
Subject Classification ◽  
Odd Order ◽  
A Finite Group

AbstractLet k be a non-zero complex number and let u and v be elements of a finite group G. Suppose that at most one of u and v belongs to O(G), the maximal normal subgroup of G of odd order. It is shown that G satisfies X(v)–X(u) = k for every complex nonprincipal irreducible character X in the principal 2-block of G, if and only if G/O(G) is isomorphic to one of the following groups: C2, PSL(2, 2n) or pΣL(2, 52a+1), where n≥2 and a ≥ 1.Subject classification (Amer. Math. Soc. (MOS) 1970): 20 C 20

HALF-ISOMORPHISMS OF MOUFANG LOOPS OF ODD ORDER

2012 ◽  
Vol 11 (05) ◽  
pp. 1250094 ◽  
Author(s):  
STEPHEN GAGOLA ◽  
MARIA DE LOURDES MERLINI GIULIANI
Keyword(s):  
Finite Group ◽  
Multiplicative Systems ◽  
Odd Order ◽  
Moufang Loops ◽  
Even Order ◽  
Carry Over ◽  
A Finite Group

A half-isomorphism φ : G → K between multiplicative systems G and K is a bijection from G onto K such that φ(ab) ∈ {φ(a)φ(b), φ(b)φ(a)} for any a, b ∈ G. It was shown by W. R. Scott [Half-homomorphisms of groups, Proc. Amer. Math. Soc. 8 (1957) 1141–1144] that if G is a group then φ is either an isomorphism or an anti-isomorphism. This is used to prove that a finite group is determined by its group determinant. Here we show that every half-isomorphism between Moufang loops of odd order is either an isomorphism or an anti-isomorphism. Such a result does not carry over to Moufang loops of even order.

scholarly journals A on simplicity criterion for finite groups

1969 ◽  
Vol 10 (3-4) ◽  
pp. 359-362
Author(s):  
Nita Bryce
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Odd Order ◽  
A Finite Group

M. Suzuki [3] has proved the following theorem. Let G be a finite group which has an involution t such that C = CG(t) ≅ SL(2, q) and q odd. Then G has an abelian odd order normal subgroup A such that G = CA and C ∩ A = 〈1〉.

scholarly journals On Schur p-Groups of odd order

2017 ◽  
Vol 16 (03) ◽  
pp. 1750045 ◽  
Author(s):  
Grigory Ryabov
Keyword(s):  
Finite Group ◽  
Odd Order ◽  
A Finite Group ◽  
Natural Way

A finite group [Formula: see text] is called a Schur group if any [Formula: see text]-ring over [Formula: see text] is associated in a natural way with a subgroup of [Formula: see text] that contains all right translations. We prove that the groups [Formula: see text], where [Formula: see text], are Schur. Modulo previously obtained results, it follows that every noncyclic Schur [Formula: see text]-group, where [Formula: see text] is an odd prime, is isomorphic to [Formula: see text] or [Formula: see text], [Formula: see text].

On the number of conjugacy classes in a finite group II

1988 ◽  
Vol 64 (1) ◽  
pp. 87-127 ◽  
Author(s):  
Antonio Vera-López ◽  
MA Concepción Larrea
Keyword(s):  
Finite Group ◽  
Conjugacy Classes ◽  
Group Ii ◽  
A Finite Group

On weakly σ-semipermutable subgroups of finite groups

2021 ◽  
Author(s):  
Venus Amjid ◽  
Muhammad Tanveer Hussain ◽  
Zhenfeng Wu
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Permutable Subgroup ◽  
A Finite Group

Let [Formula: see text] be some partition of the set of all primes [Formula: see text], [Formula: see text] be a finite group and [Formula: see text]. A set [Formula: see text] of subgroups of [Formula: see text] is said to be a complete Hall[Formula: see text]-set of [Formula: see text] if every non-identity member of [Formula: see text] is a Hall [Formula: see text]-subgroup of [Formula: see text] for some [Formula: see text] and [Formula: see text] contains exactly one Hall [Formula: see text]-subgroup of [Formula: see text] for every [Formula: see text]. Let [Formula: see text] be a complete Hall [Formula: see text]-set of [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-semipermutable with respect to [Formula: see text] if [Formula: see text] for all [Formula: see text] and all [Formula: see text] such that [Formula: see text]; [Formula: see text]-semipermutablein [Formula: see text] if [Formula: see text] is [Formula: see text]-semipermutable in [Formula: see text] with respect to some complete Hall [Formula: see text]-set of [Formula: see text]. We say that a subgroup [Formula: see text] of [Formula: see text] is weakly[Formula: see text]-semipermutable in [Formula: see text] if there exists a [Formula: see text]-permutable subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] is [Formula: see text]-permutable in [Formula: see text] and [Formula: see text], where [Formula: see text] is the subgroup of [Formula: see text] generated by all those subgroups of [Formula: see text] which are [Formula: see text]-semipermutable in [Formula: see text]. In this paper, we study the structure of [Formula: see text] under the condition that some subgroups of [Formula: see text] are weakly [Formula: see text]-semipermutable in [Formula: see text].

Subgroups of Central Separable Algebras

1973 ◽  
Vol 25 (4) ◽  
pp. 881-887 ◽  
Author(s):  
E. D. Elgethun
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Division Ring ◽  
Multiplicative Group ◽  
Prime Field ◽  
Odd Order ◽  
Multiplicative Subgroup ◽  
A Finite Group ◽  
Separable Algebras

In [8] I. N. Herstein conjectured that all the finite odd order sub-groups of the multiplicative group in a division ring are cyclic. This conjecture was proved false in general by S. A. Amitsur in [1]. In his paper Amitsur classifies all finite groups which can appear as a multiplicative subgroup of a division ring. Let D be a division ring with prime field k and let G be a finite group isomorphic to a multiplicative subgroup of D.

scholarly journals Abelian Sylow subgroups in a finite group, II

2015 ◽  
Vol 421 ◽  
pp. 3-11 ◽  
Author(s):  
Gabriel Navarro ◽  
Ronald Solomon ◽  
Pham Huu Tiep
Keyword(s):  
Finite Group ◽  
Sylow Subgroups ◽  
Group Ii ◽  
A Finite Group
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