Automorphisms which centralize a Sylow p-subgroup

Gross, Fletcher

doi:10.1016/0021-8693(82)90287-3

Cited by 48 publications

(23 citation statements)

References 3 publications

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“…The last argument in the proof, which will be used few times more, has been used already by F. Gross [13,Corollary 2.4] to prove the following Finally, the following remark will prove to be very useful, and will be applied throughout the paper.…”

Section: Lemmamentioning

confidence: 99%

Coleman automorphisms of finite groups

Hertweck

Kimmerle

2002

Mathematische Zeitschrift

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An automorphism σ of a finite group G whose restriction to any Sylow subgroup equals the restriction of some inner automorphism of G shall be called Coleman automorphism, named for D. B. Coleman, who's important observation from [2] especially shows that such automorphisms occur naturally in the study of the normalizer N of G in the units U of the integral group ZG. Let Out Col (G) be the image of these automorphisms in Out(G). We prove that Out Col (G) is always an abelian group (based on previous work of E. C. Dade, who showed that Out Col (G) is always nilpotent). We prove that if no composition factor of G has order p (a fixed prime), then Out Col (G) is a p -group. If O p (G) = 1, it suffices to assume that no chief factor of G has order p. If G is solvable and no chief factor of G/O 2 (G) has order 2, then N = GZ, where Z is the center of U. This improves an earlier result of S.

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Section: Lemmamentioning

confidence: 99%

Coleman automorphisms of finite groups

Hertweck

Kimmerle

2002

Mathematische Zeitschrift

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“…. .AThe following proposition is based on this lemma and an observation of F. Gross[4, Corollary 2.4]. It shows that a`conjugating unit' occurring in the F Ã -Theorem is unique up to central units and group elements.…”

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confidence: 92%

On Principal Blocks of p -Constrained Groups

Hertweck

Kimmerle

2002

Proceedings of the London Mathematical Society

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A theorem of K. W. Roggenkamp and L. L. Scott shows that for a finite group G with a normal p‐subgroup containing its own centralizer, any two group bases of the integral group ring ZG are conjugate in the units of ZpG. Though the theorem presents itself in the work of others and appears to be needed, there is no published account. There seems to be a flaw in the proof, because a ‘theorem’ appearing in the survey [K. W. Roggenkamp, ‘The isomorphism problem for integral group rings of finite groups’, Progress in Mathematics 95 (Birkhäuser, Basel, 1991), pp. 193‐‐220], where the main ingredients of a proof are given, is false. In this paper, it is shown how to close this gap, at least if one is only interested in the conclusion mentioned above. Therefore, some consequences of the results of A. Weiss on permutation modules are stated. The basic steps of which any proof should consist are discussed in some detail. In doing so, a complete, yet short, proof of the theorem is given in the case that G has a normal Sylow p‐subgroup. 2000 Mathematical Subject Classification: primary 16U60; secondary 20C05.

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“…Our approach does not however use the theory of normal maps (see [4, § 7]). Instead we rely on a group theoretic result, which is a corollary to the following theorem of Gross in [11]: Then C has a normal p-complement.…”

Section: Existence Of C F (E)mentioning

confidence: 99%