On Groups of Even Order

Brauer, Richard; Fowler, K. A.

doi:10.2307/1970080

Cited by 226 publications

(181 citation statements)

References 2 publications

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“…From the definition it is clear that Cg(o)QCq(o-). Hence the condition (1') is a consequence of the condition (1). Suppose that (T = tt'.…”

Section: Theoremmentioning

confidence: 99%

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Finite groups with nilpotent centralizers

Suzuki¹

1961

Trans. Amer. Math. Soc.

116

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“…From the definition it is clear that Cg(o)QCq(o-). Hence the condition (1') is a consequence of the condition (1). Suppose that (T = tt'.…”

Section: Theoremmentioning

confidence: 99%

“…If § is not empty we can take a subgroup H which belongs to ^V Then by (2) the normalizer N=Ng(H) contains at least two Sylow 2-groupsIf Q and Q' are Sylow 2-groups of N, QC\Q' contains H and H^e by (1). By a theorem of Sylow Q and Q' are contained in Sylow 2-groups P and P' of G respectively.…”

Section: Propositionmentioning

confidence: 99%

Finite groups with nilpotent centralizers

Suzuki¹

1961

Trans. Amer. Math. Soc.

116

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“…For an excellent survey on the Hamiltonian cycles in Cayley graphs see [17]. Aside this, several other graphs have also been associated with finite groups such as commuting graphs, intersection graphs, prime graphs, non-commuting graphs, conjugacy class graphs, etc, [5,6,11,12,18]. In order to justify our claim that the inverse graph is new, we as well illustrate by examples how it is different from some known graphs associated with groups.…”

Section: Introductionmentioning

confidence: 98%

Inverse graphs associated with finite groups

Alfuraidan

Zakariya

2017

ejgta

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“…From a property of 2-groups 5 contains an involution r contained in the center of 5. Then the centralizer of r contains 5 and Z, and is abelian by the assumption (1). Hence both S and Z are abelian.…”

mentioning

confidence: 93%

“…Let G be a finite group, 5 a 2-Sylow subgroup, Z the centralizer of S and N the normalizer of 5 in G. Throughout this section we assume that (1) The centralizer of any involution in G is abelian. Under this assumption we have the following proposition.…”

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confidence: 99%

Untitled

1959

Trans. Amer. Math. Soc.

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Let Lq be the totality of nonsingular 2X2 matrices with determinant 1 over a finite field of q elements. This paper is concerned with a grouptheoretical characterization of Lq in the case that q is a power of 2. We shall call an element of order 2 an involution. It is easy to see that Lq (q = 2n) contains an involution t such that the centralizer of r in Lq is an abelian 2-group. In his thesis ). The purpose of this paper is to prove a further generalization of these results: i.e. to prove the following theorem.Main Theorem. Let G be a finite group of even order. If the centralizer of any involution in G is always abelian then we have one of the following three possibilities: (1) 2-Sylow subgroups of G are cyclic, (2) a 2-Sylow subgroup of G is a normal subgroup, or (3) G is a direct product of two groups L and A where L is one of the linear groups Lq with q = 2" and A is an abelian group of odd order.The proof of this result is rather complicated.The first section is devoted to a reduction process. Assuming our main theorem to be false we study the structure of the group with the smallest order for which the assertion of the main theorem is not true. The second section is concerned with a particular type of finite groups with structure similar to the one obtained at the end of the first section. In view of further applications we take slightly more general assumptions. Our main purpose is to obtain a formula for the order of such a group which is applied in the next section to prove the nonexistence of the group considered in §1. The method used here is similar to the one in [l] or [6]. An application of our main theorem will be given in the final section.

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On Groups of Even Order

Cited by 226 publications

References 2 publications

Finite groups with nilpotent centralizers

Finite groups with nilpotent centralizers

Inverse graphs associated with finite groups

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