Normalizers of 2-subgroups of finite groups (original) (raw)

2004

Using the classification of finite simple groups we prove the following statement: Let p > 3 be a prime, Q a group of automorphisms of p-power order of a finite group G, and P a Q-invariant Sylow p-subgroup of G. If C N G (P)/P (Q) is trivial, then G is solvable. An equivalent formulation is that if G has a self-normalizing Sylow p-subgroup with p > 3 a prime, then G is solvable. We also investigate the possibilities when p = 3. Theorem 1.1. Let p be an odd prime and P a Sylow p-subgroup of the finite group G. If p = 3, assume that G has no composition factors of type L 2 (3 f), f = 3 a with a ≥ 1. (1) If P = N G (P), then G is solvable. (2) If N G (P) = P C G (P), then G/O p (G) is solvable. Note that the second result implies the first since it is well known ([7], [1, Lemma 12.1]) that if H is a group of automorphisms of R with gcd(|H|, |R|) = 1 and C R (H) = 1, then R is solvable. We then apply this result to P acting on O p (G). We will say more about this in the next section. If G is a simple group with p ≥ 5, it was an observation of Thompson that this followed quite easily from a result of Glauberman. See [4, Thm. X.8.15]. An easy consequence of the previous theorem (or our proof) is the extension of this result to p = 3. Corollary 1.2. If p is an odd prime and G is a nonabelian finite simple group, then N G (P) = P C G (P). Proof. By the theorem, we need only consider p = 3 and G = L 2 (3 3 a). Then the split torus acts nontrivially on a Sylow 3-subgroup.

A characterization of the simple group He

Journal of Algebra, 1979

The purpose of this paper is to prove the THEOREM. Let G be a jinite, noaz-abelian, simple group containing a 3-central element r of order 3 such that C,(rr)j(w> is isomorphic to A,. If G is not 3-~o~~a~ then G is isomorphic to He. This yields the following COKQLLARY. Let G be a jkite, non-abelian, sim@e group containing alz element T of order 3 such that Co(~))/(~) is isomorphic to A, or S,. Assume f~~~the~mQ~e at there exists in G an elementary abelian subgroup of order 9 all rzon-identity elements of which me conjugate to rr. Then G is isomorphic to He. The proof of the Theorem will be based on the following YPOTI-IESIS. Let G be a simple group containing a standard subgroup A stich A/Z(A) is isomorphic to PSL(3, 4). Then G : ZF i.~omoqphic to He or SW. OY Ol$r~ Here He, Suz, ON denote the sporadic simple groups of orders 1P33527~17, 21337527.1H "13, 2g345721 1.19.31 respectively discovered by O'Nan. In [3] Gheng proves the Hypothesis under one of the following further conditions: (i) The Sylow 2-subgroup of Z(A) is non-trivial. (ii) 211 does not divide the order of G. The full result might be established by now. The notation is standard (see [q and [lo]). In particular A, and S, denote respectively the alternating and symmetric group of degree n. PSL(n, q) is the projective special linear group of dimension n on a field with p elements. Z, denotes the cyclic group of order n and E,n the elementary abelian p-group of 261

On finite factorizable groups*1

Journal of Algebra, 1984

ON FINITE FACTORIZABLE GROUPS 523 (I) A, with r > 5 a prime and A N A,-, . (II) M,, and either A is solvable or A N M,,. (III) M,, and either B is Frobenius of order 11 . 23 or B is cyclic of order 23 and A N M,, .

On pointwise inner automorphisms of nilpotent groups of class2

2013

An automorphism of a group G is pointwise inner if is conjugate to for any . The set of all pointwise inner automorphisms of group G, denoted by form a subgroups of containing . In this paper, we find a necessary and sufficient condition in certain finitely generated nilpotent groups of class 2 for which . We also prove that in a nilpotent group of class 2 with cyclic commutator subgroup and the quotient is torsion. In particular if is a finite cyclic group then .

Normalizers of 2-subgroups of finite groups (original) (raw)

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