Carter Subgroups in Classical Groups

Martino, L. Di; Tamburini, M. C.; Zalesskiǐ, A. E.

doi:10.1112/s0024610797004869

Cited by 11 publications

(17 citation statements)

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“…Indeed, the existence of Carter subgroups in (a) and (c) follows from there being a Carter subgroup of order 6 in PGU 3 (2) (see [5]). Throughout (b), (d)-(f), such groups exist by reason of the fact that a Sylow 2-subgroup in a group of Lie type defined over a field of order 2 coincides with its normalizer.…”

Section: Now Assume Thatmentioning

confidence: 94%

“…2. 5.13], we call ζ a field automorphism if ε = 0, that is, ζ = ϕ , and call it a graph-field automorphism in all other cases (under the assumption that ϕ = e). field automorphism if |ζ| is not divisible by |γ| (this definition will also be used in the case where |γ| = 3 and G σ 3 D 4 (q 3 )), and call it a graph automorphism in all other cases.…”

Section: Semilinear Groups Of Lie Typementioning

confidence: 99%

“…In favor of this conjecture is evidence coming from extensive studies of classes of finite groups that are close to simple. In particular, it was shown that the conjecture holds true for symmetric and alternating groups [2]; for any group A such that SL n (p t ) ≤ A ≤ GL n (p t ) [3,4]; for symplectic groups Sp 2n (p t ), full unitary groups GU n (p 2t ), and full orthogonal groups GO ± n (p t ) where p is odd [5] (p t is a power of a prime p). In [6] the results of [5] were extended to any group G with O p (S) ≤ G ≤ S, where S is a full classical matrix group.…”

Section: Preliminariesmentioning

confidence: 99%

“…Since −1 is not a square in the base field of G (i.e., in GF (q)), we have q ≡ −1 (mod 4), whence h s (−1) | s ∈ Φ = Q. P LEMMA 2. 5. Let G = P Sp 2n (q) be a simple canonical group of Lie type, J be a subset of the set of fundamental roots containing r n as a long fundamental root, P J be a parabolic subgroup generated by the Borel subgroup B and by groups X r with −r ∈ J, and L be a Levi factor of P J .…”

Section: Lemma 22 (Hartley-shute Lemma)mentioning

confidence: 99%

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Carter subgroups of finite almost simple groups

Vdovin

2007

Algebr Logic

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In the paper we complete the classification of Carter subgroups in finite almost simple groups. In particular, we prove that Carter subgroups of every finite almost simple group are conjugate. Togeather with previous results by author and F. Dalla Volta, A. Lucchini, and M. C. Tamburini, as a corollary, it follows that Carter subgroups of every finite group are conjugate.Lemma 2.3. Let G be a finite group and S be a Sylow 2-subgroup of G. Then G contains a Carter subgroup K with S ≤ K if and only if N G (S) = SC G (S).Proof. Assume that G contains a Carter subgroup K with S ≤ K. Since K is nilpotent, it follows that S is normal in K and K ≤ SC G (S) ✂ N G (S). By Feit-Thompson Theorem (see [11]) we obtain that N G (S) is solvable. Thus, by Lemma 2.2(1) we have that SC) is of odd order, it is solvable. Hence it contains a Carter subgroup K 1 . Consider a nilpotent subgroup K = S × K 1 of G. Assume that x ∈ N G (K), then x ∈ N G (S). But K is a Carter subgroup of N G (S), hence x ∈ K and K is a Carter subgroup of G.

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Section: Now Assume Thatmentioning

confidence: 94%

Section: Semilinear Groups Of Lie Typementioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Lemma 22 (Hartley-shute Lemma)mentioning

confidence: 99%

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Carter subgroups of finite almost simple groups

Vdovin

2007

Algebr Logic

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“…In addition to the references already mentioned we quote [26], [8], [12], [22]. And, above all, the recent paper of E.P.…”

Section: Introductionmentioning

confidence: 99%

The Conjugacy Conjecture for Carter Subgroups

Tamburini

2007

Milan j. math.

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This paper concerns the problem of determining the nilpotent self-normalizing (Carter) subgroups of a finite group, in relation to the conjecture that they are conjugate, when they exist. We outline the origin of this conjecture and the methods by which it has been studied. For the most substantial contribution, we refer to the recent paper of E.P. Vdovin [27]. As an illustration of the methods, we determine the Carter subgroups of the almost simple groups with socle PSL n (q), in characteristic r 0 ≥ 5.

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Finite Groups over Arithmetical Rings and Globally Irreducible Representations

Oystaeyen

1999

Journal of Algebra

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Given the ring of integers R of an algebraic number field K, for which natural Ž . number n is there a finite group G ; GL n, R such that RG, the R-span of G, Ž . Ž . Ž . coincides with M n, R , the ring of n = n -matrices over R? Given G ; GL n, R Ž . we show that RG s M n, R if and only if the Brauer reduction of G modulo every prime is absolutely irreducible. In addition, the question above is fully answered if n is an odd prime. ᮊ

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Carter Subgroups in Classical Groups

Cited by 11 publications

References 0 publications

Carter subgroups of finite almost simple groups

Carter subgroups of finite almost simple groups

The Conjugacy Conjecture for Carter Subgroups

Finite Groups over Arithmetical Rings and Globally Irreducible Representations

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