gabriel ihuitl navarro | Universidad Autonoma Gabriel Rene Moreno (original) (raw)
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Using the classification of finite simple groups we prove the following statement: Let p > 3 be a... more 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.
Using the classification of finite simple groups we prove the following statement: Let p > 3 be a... more 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.