Coleman Automorphisms of Extensions of Finite Characteristically Simple Groups by Some Finite Groups

2021 ◽  
Vol 28 (04) ◽  
pp. 561-568
Author(s):  
Jinke Hai ◽  
Lele Zhao
Keyword(s):  
Abelian Group ◽  
Simple Group ◽  
Finite Groups ◽  
Finite Simple Group ◽  
Simple Groups ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Inner Automorphism ◽  
Coleman Automorphism

Let [Formula: see text] be an extension of a finite characteristically simple group by an abelian group or a finite simple group. It is shown that every Coleman automorphism of [Formula: see text] is an inner automorphism. Interest in such automorphisms arises from the study of the normalizer problem for integral group rings.

On Coleman Automorphisms of Extensions of Finite Quasinilpotent Groups by Some Groups

2018 ◽  
Vol 25 (02) ◽  
pp. 181-188 ◽  
Author(s):  
Jinke Hai ◽  
Yixin Zhu
Keyword(s):  
Finite Group ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Inner Automorphism ◽  
Coleman Automorphism ◽  
A Finite Group

Let G be an extension of a finite quasinilpotent group by a finite group. It is shown that under some conditions every Coleman automorphism of G is an inner automorphism. The interest in such automorphisms arose from the study of the normalizer problem for integral group rings. Our theorems generalize some well-known results.

PYTHAGOREAN TRIPLES AND UNITS IN INTEGRAL GROUP RINGS

2005 ◽  
Vol 04 (04) ◽  
pp. 355-367 ◽  
Author(s):  
M. BEATTIE ◽  
C. WEATHERBY
Keyword(s):  
Abelian Group ◽  
Equivalence Relation ◽  
Dihedral Group ◽  
Group Rings ◽  
Integral Group ◽  
Simple Method ◽  
Integer Points ◽  
Integral Group Rings ◽  
Pythagorean Triples ◽  
Inner Automorphism

In this note, we describe a simple method for finding units of group rings of the form Z[G] = Z[H]#Z[C2] for H an abelian group, and apply this to the case G = D4, the dihedral group of order 8. Here, units may be described as integer points on hyperboloids, and, defining units u, v to be equivalent if they differ by an inner automorphism, we see that this equivalence relation partitions each hyperboloid into finitely many classes.

On the Isomorphism of Integral Group Rings. I

1969 ◽  
Vol 21 ◽  
pp. 410-413 ◽  
Author(s):  
Sudarshan K. Sehgal
Keyword(s):  
Finite Groups ◽  
Classical Result ◽  
Abelian Groups ◽  
Group Rings ◽  
Integral Group ◽  
Normal Subgroups ◽  
Finite Abelian Groups ◽  
Integral Group Rings ◽  
Inner Automorphism ◽  
A Finite Group

In this note we study the question of automorphisms of the integral group ring Z(G) of a finite group G. We prove that if G is nilpotent of class two, any automorphism of Z(G) is composed of an automorphism of G and an inner automorphism by a suitable unit of Q(G), the group algebra of G with rational coefficients. In § 3, we prove that if two finitely generated abelian groups have isomorphic integral group rings, then the groups are isomorphic. This is an extension of the classical result of Higman (2) for the case of finite abelian groups. In the last section we give a new proof of the fact that an isomorphism of integral group rings of finite groups preserves the lattice of normal subgroups. Other proofs are given in (1;4).

scholarly journals On the Gruenberg–Kegel graph of integral group rings of finite groups

2017 ◽  
Vol 27 (06) ◽  
pp. 619-631 ◽  
Author(s):  
W. Kimmerle ◽  
A. Konovalov
Keyword(s):  
Group Ring ◽  
Finite Groups ◽  
Prime Graph ◽  
Unit Group ◽  
Integral Group Ring ◽  
Simple Groups ◽  
Group Rings ◽  
Integral Group ◽  
Almost Simple Groups ◽  
Integral Group Rings

The prime graph question asks whether the Gruenberg–Kegel graph of an integral group ring [Formula: see text], i.e. the prime graph of the normalized unit group of [Formula: see text], coincides with that one of the group [Formula: see text]. In this note, we prove for finite groups [Formula: see text] a reduction of the prime graph question to almost simple groups. We apply this reduction to finite groups [Formula: see text] whose order is divisible by at most three primes and show that the Gruenberg–Kegel graph of such groups coincides with the prime graph of [Formula: see text].

scholarly journals On the Structure of Integral Group Rings of Sporadic Groups

2000 ◽  
Vol 3 ◽  
pp. 274-306 ◽  
Author(s):  
Frauke M. Bleher ◽  
Wolfgang Kimmerle
Keyword(s):  
Automorphism Groups ◽  
Symmetric Groups ◽  
Computational Procedure ◽  
Isomorphism Problem ◽  
Simple Groups ◽  
Group Rings ◽  
Integral Group ◽  
Sporadic Groups ◽  
Integral Group Rings ◽  
Sporadic Simple Groups

AbstractThe object of this article is to examine a conjecture of Zassenhaus and certain variations of it for integral group rings of sporadic groups. We prove the ℚ-variation and the Sylow variation for all sporadic groups and their automorphism groups. The Zassenhaus conjecture is established for eighteen of the sporadic simple groups, and for all automorphism groups of sporadic simple groups G which are different from G. The proofs are given with the aid of the GAP computer algebra program by applying a computational procedure to the ordinary and modular character tables of the groups. It is also shown that the isomorphism problem of integral group rings has a positive answer for certain almost simple groups, in particular for the double covers of the symmetric groups.

On the number of Sylow subgroups in finite simple groups

2020 ◽  
pp. 2150115
Author(s):  
Zhenfeng Wu
Keyword(s):  
Group Theory ◽  
Simple Group ◽  
Finite Groups ◽  
Finite Simple Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Sylow Subgroups

Denote by [Formula: see text] the number of Sylow [Formula: see text]-subgroups of [Formula: see text]. For every subgroup [Formula: see text] of [Formula: see text], it is easy to see that [Formula: see text], but [Formula: see text] does not divide [Formula: see text] in general. Following [W. Guo and E. P. Vdovin, Number of Sylow subgroups in finite groups, J. Group Theory 21(4) (2018) 695–712], we say that a group [Formula: see text] satisfies DivSyl(p) if [Formula: see text] divides [Formula: see text] for every subgroup [Formula: see text] of [Formula: see text]. In this paper, we show that “almost for every” finite simple group [Formula: see text], there exists a prime [Formula: see text] such that [Formula: see text] does not satisfy DivSyl(p).

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