Finite groups with odd Sylow normalizers (original) (raw)

Journal of Algebra and Its Applications, 2014

The paper considers the influence of Sylow normalizers, i.e., normalizers of Sylow subgroups, on the structure of finite groups. In the universe of finite soluble groups it is known that classes of groups with nilpotent Hall subgroups for given sets of primes are exactly the subgroup-closed saturated formations satisfying the following property: a group belong to the class if and only if its Sylow normalizers do so. The paper analyzes the extension of this research to the universe of all finite groups.

On Sylow Numbers of Some Finite Groups

we will show that if |G| = |S| and NS(G) = NS(S), where S is one of the groups: the special projective linear groups L3(q), with 5 (q −1), the projective special unitary groups U3(q), the sporadic simple groups, the alternating simple groups, and the symmetric groups of degree prime r, then G is isomorphic to S. Furthermore, we will show that if G is a finite centerless group and NS(G) = NS(L2(17)), then G is isomorphic to L2(17), and or G is isomorphic to Aut(L2(17).

Self-normalizing Sylow subgroups

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 Characterisation of Certain Finite Groups of Odd Order

Mathematical Proceedings of the Royal Irish Academy, 2011

The commutativity degree of a finite group is the probability that two randomly chosen group elements commute. The main object of this paper is to obtain a characterization for all finite groups of odd order with commutativity degree greater than or equal to 11 75 .

Groups with the same orders of Sylow normalizers as the Mathieu groups

International Journal of Mathematics and Mathematical Sciences, 2005

There exist many characterizations for the sporadic simple groups. In this paper we give two new characterizations for the Mathieu sporadic groups. Let M be a Mathieu group and let p be the greatest prime divisor of |M|. In this paper, we prove that M is uniquely determined by |M| and |N M (P)|, where P ∈ Syl p (M). Also we prove that if G is a finite group, then G ∼ = M if and only if for every prime q,

On the order of the Sylow subgroups of the automorphism group of a finite group

Glasgow Mathematical Journal, 1970

1. Introduction. Given any finite group G, we wish to determine a relationship between the highest power of a prime p dividing the order of G, denoted by | G \ p , and | A{G) \ p , where A{G) is the automorphism group of G. It was shown by Herstein and Adney [8] that | A(G) \ p ŵ henever | G \ p ^ p 2. Later Scott [16] showed that | A(G) special case where G is abelian, Hilton [9] proved that Adney [1] showed that this result holds if a Sylow /^-subgroup of G is abelian, and gave an example where G\ p =/>* and |/4(G)| P =p 2. We are able to show in Theorem 4.5 that, if | G\ p ^p s , then A(G)\ p^p 3. In the general case, Ledermann and Neumann [11] showed that there exists a function gift) having the property that | A(G) \ p ^ p h whenever | G \ p ^ p 9(h) , and gave an upper bound for g(h). Later, Green [6] improved their result by showing that " ^ p 2 whenever | G \ p ^ p 3. For the A(G) \ p ^ p"~' whenever | G \ p ^ p". Howarth [10] then proved that, for h ^ 12,f Ji(/i 2 + 3) for h odd, 9 ~~W» 2 +4) for h even. We are able to improve this result by showing that, for all h, We shall also consider the special case where G is a p-group, and show that in this case | A(G) \ p ^ p h whenever | G \ ;> p A , where tt(+) for h\ h + l for We point out that all groups considered in this paper are finite. Also, the letter p will always stand for a prime. 2. Central automorphisms. An automorphism a such that g~1g" is in the center of G, for all g in G, is called central. The set of all central automorphisms of G forms a subgroup of A(G), which we denote by A C (G). It is easy to show that A C (G) is the centralizer of the inner automorphism group I(G) in A(G). From this it follows that A C (G) is normal in A(G), and that t Howarth remarks that the result can be shown to be valid for h 1 6.

A solvability criterion for finite groups related to the number of Sylow subgroups

Communications in Algebra, 2020

Let G be a finite group and let pðGÞ be the set of primes dividing the order of G. For each p 2 pðGÞ, the Sylow theorems state that the number of Sylow p-subgroups of G is equal to kp þ 1 for some non-negative integer k. In this article, we characterize non-solvable groups G containing at most p 2 þ 1 Sylow p-subgroups for each p 2 pðGÞ: In particular, we show that each finite group G containing at most ðp À 1Þp þ 1 Sylow p-subgroups for each p 2 pðGÞ is solvable.

Groups with the same orders of Sylow normalizers as the Janko groups

J. Appl. Algebra Discrete Struct, 2005

MATH 436 Notes: Sylow Theory

2010

We are now ready to apply the theory of group actions we studied in the last section to study the general structure of finite groups. A key role is played by the p-subgroups of a group. We will see that the Sylow theory will give us a way to study a group “a prime at a time”. First we record a very important special case of group actions: Theorem 1.1 (p-group Actions). Let p be a prime. If P is a p-group acting on a finite set X then |X| ≡ |X | mod p. Proof. By the orbit decomposition formula we have: |X| = |X | + ∑

Finite groups with odd Sylow normalizers (original) (raw)

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