The Normalizer of a Finite Group in its Integral Group Ring and Čech Cohomology

In this paper the normalizer problem of an integral group ring of an arbitrary group G is investigated. It is shown that any element of the normalizer 1 G of G in the group of normalized units 1 G is determined by a finite normal subgroup. This reduction to finite normal subgroups implies that the normalizer property holds for many classes of (infinite) groups, such as groups without non-trivial 2-torsion, torsion groups with a normal Sylow 2-subgroup, and locally nilpotent groups. Further it is shown that the commutator of 1 G equals G and 1 G /G is finitely generated if the torsion subgroup of the finite conjugacy group of G is finite.  2002 Elsevier Science

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“…(5) (Marciniak and Roggenkamp [12] and Mazur [15]) If u ∈ 1 G and u n ∈ G for some positive integer n, then u ∈ G.…”

Section: Reduction To Finite Groupsmentioning

confidence: 99%

“…This normalizer property (NP) has been proved in the affirmative for many classes of finite groups [1,3,5,8,9,11,12,16]. For infinite groups much less is known: Mazur [15] proved (NP) for any torsion free group and in [11] some special classes of infinite groups were dealt with.…”

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confidence: 95%

On the Normalizer Problem

et al. 2002

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“…Recall that an automorphism ρ of a finite group G is called a Coleman automorphism, provided that ρ 2 ∈ Inn(G), ρ preserves the conjugacy classes of G and the restriction of ρ to any Sylow subgroup of G equals the restriction of some inner automorphism of G. This definition was initially introduced by Marciniak and Roggenkamp [14]. All Coleman automorphisms of G form a group, denoted by Aut C (G).…”

Section: Introductionmentioning

confidence: 99%

On Coleman automorphisms of wreath products of finite nilpotent groups by abelian groups

Hai

2011

Sci. China Math.

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Let G = NwrA be a wreath product of a finite nilpotent group N by an abelian group A. It is shown that every Coleman automorphism of G is an inner automorphism. As an immediate consequence of this result, it is obtained that the normalizer property holds for G. KeywordsColeman automorphisms, wreath products, the normalizer problem MSC(2000): 20C05, 16S34, 20C10Citation: Hai J K, Li Z X. On Coleman automorphisms of wreath products of finite nilpotent groups by abelian groups.

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“…Thus it is no surprise that Coleman automorphisms of G occur naturally in the study of the normalizer problem. Related work in this direction can be found in [3][4][5][6][7][8][9][10][11][12]. However, we do not intend to go into details for this.…”

mentioning

confidence: 95%

Coleman automorphisms of generalized dihedral groups

2016

Acta. Math. Sin.-English Ser.

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Let D be a generalized dihedral group and Aut Col (D) its Coleman automorphism group. Denote by Out Col (D) the quotient group of Aut Col (D) by Inn(D), where Inn(D) is the inner automorphism group of D. It is proved that either Out Col (D) = 1 or Out Col (D) is an elementary abelian 2-group whose order is completely determined by the cardinality of π(D). Furthermore, a necessary and sufficient condition for Out Col (D) = 1 is obtained. In addition, whenever Out Col (D) = 1, it is proved that Aut Col (D) is a split extension of Inn(D) by an elementary abelian 2-group for which an explicit description is given.

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The Normalizer of a Finite Group in its Integral Group Ring and Čech Cohomology

Cited by 25 publications

References 6 publications

On the Normalizer Problem

On the Normalizer Problem

On Coleman automorphisms of wreath products of finite nilpotent groups by abelian groups

Coleman automorphisms of generalized dihedral groups

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