A Characterization of the Zassenhaus Groups

Harada, Koichiro

doi:10.1017/s0027763000012897

Cited by 8 publications

(10 citation statements)

References 12 publications

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“…Our notations are taken from [5] and character notation from [6]. If A" is a nonempty set, let \K\ denote the number of elements in K. If p is a prime, then Gp denotes a Sylow p subgroup of G.…”

Section: Corollarymentioning

confidence: 99%

“…Hence, there are /, irreducible characters X¡, /= 1, ...,/,, of G which are associated with Ax and one nonprincipal irreducible character Y such that Y(z) ¥= 0 for z E A\ [6]. Let 1G denote the principal character of G. Lemma 3 [6] implies ^,(1) = knx + 2e and Y(l) = knx + e where A: is a positive integer and e = ±1. Lemma 6 [6] implies |G| = nxXx(l)Y(\)(lnx + 1) where / is a nonnegative integer.…”

Section: Corollarymentioning

confidence: 99%

“…Let 1G denote the principal character of G. Lemma 3 [6] implies ^,(1) = knx + 2e and Y(l) = knx + e where A: is a positive integer and e = ±1. Lemma 6 [6] implies |G| = nxXx(l)Y(\)(lnx + 1) where / is a nonnegative integer.…”

Section: Corollarymentioning

confidence: 99%

“…(See [6], [7] and [8].) It is an open question of Higman [7] whether or not there are finite simple groups of type (2,4,4) other than the Suzuki groups.…”

mentioning

confidence: 99%

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On finite groups containing three 𝐶𝐶-subgroups

Arad¹,

Ferguson²

1980

Proc. Amer. Math. Soc.

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Section: Corollarymentioning

confidence: 99%

Section: Corollarymentioning

confidence: 99%

Section: Corollarymentioning

confidence: 99%

“…(See [6], [7] and [8].) It is an open question of Higman [7] whether or not there are finite simple groups of type (2,4,4) other than the Suzuki groups.…”

mentioning

confidence: 99%

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On finite groups containing three 𝐶𝐶-subgroups

Arad¹,

Ferguson²

1980

Proc. Amer. Math. Soc.

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“…However ^,(1) is even for PSL (3,4) or PSL(2, q), q odd. Let G2 be a Sylow 2 subgroup of G ; if G2 is abelian, then [7] again implies a contradiction.…”

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confidence: 99%

On finite simple groups with a self-centralization system of type (2(𝑛))

Ferguson¹

1978

Proc. Amer. Math. Soc.

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Abstract. Let G denote a simple group with a self-centralization system of type (2(n)), where n > 3. Let X", denote an exceptional character of G, then Xx(l) = kn + 2e where e = ± 1. It is known that \G\-^,(1X^,(1) -e)(*i+ 1) where / is a nonnegative integer. In this paper G is classified if / = 0, e = 1 and X"|(l) is odd.Let G be a finite simple group, a proper subgroup A of G is called a CC subgroup if CG(a) C A for all a E A*. If |tfc(¿)|/|¿| = 2 and \A\ = n, then G is said to have a self-centralization system of type (2(h)). The classification of simple groups with a self-centralization system of type (2(n)) is still incomplete. If n = 3, then G s PSL(2, q), q = 5 or 7 [2]. If n > 3, it is well known [5] that G has (n -l)/2 irreducible characters X" and one nonprincipal irreducible character Y such that Xx(l) = kn + 2e, Y(l) = kn + e where e = ± 1 and \G\ = nXx(l)Y(l)(ln + 1) where / is a nonnegative integer. In all the known simple groups of type (2(n)), 1 = 0 [5]. In this paper we classify all simple groups G with a self-centralization system of type (2(n)) where / = 0, e = 1, and ^,(1) is odd. In particular we prove the following: Theorem A. Let G be a finite simple group with a self-centralization system of type (2(n)) where n > 5. Let A be a subgroup of order n and let Xx be an exceptional character of G associated with N(A). Assume Xx(l) = kn + 2, ,(1) is odd and \G\ = n(kn + 2)(kn + 1), then G is isomorphic to Sz(q) or PSL(2, 2r).Let G E Hypothesis A if G satisfies the hypothesis of Theorem A but not the conclusion. Let r be an involution in N(A), Theorem 5.1 [5] implies G has one class of involutions. If S is a set, let |5| denote the number of elements in S.Assume G G Hypothesis A.If |Cc(t)| = 2r, then [6] implies either G satisfies Theorem A, G s PSL(3, 4) or G s PSL(2, q) where q is odd, n = (q + l)/2 or (q -l)/2 where n is odd. However ^,(1) is even for PSL(3, 4) or PSL(2, q), q odd. Let G2 be a Sylow 2 subgroup of G ; if G2 is abelian, then [7] again implies a

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Finite Projective Planes and Sharply 2-Transitive Subsets of Finite Groups

Lorimer¹

1974

Lecture Notes in Mathematics

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A Characterization of the Zassenhaus Groups

Abstract: A doubly transitive permutation group on the set of symbols Ω is called a Zassenhaus group if satisfies the following condition: the identity is the only element leaving three distinct symbols fixed.

Cited by 8 publications

References 12 publications

On finite groups containing three 𝐶𝐶-subgroups

On finite groups containing three 𝐶𝐶-subgroups

On finite simple groups with a self-centralization system of type (2(𝑛))

Finite Projective Planes and Sharply 2-Transitive Subsets of Finite Groups

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