On the p-length of some finite p-soluble groups (original) (raw)
Related papers
The nilpotent length of finite soluble groups
Journal of the Australian Mathematical Society, 1973
References Chapter 1. Preliininaries 1.1. General results. The Frattini subgroup of a group G, denoted by <I'(G), is the intersection of all the maximal subgroups of G. An element ^G is said to be omissible in G if, whenever <g,X> = G for some subset X of G, then G =<x). We now state som.e well-known results concerning the Frattini subgroup, v;hich will be used frequently in the sequel. The proofs of these results can all be found in Section 3.3 of [7]. Lemma 1.1.1 (a) An element x of G lies in ^ (.G) if and only if x is omissible in G. (b) If N G then G has a proper subgroup H such that NH = G if and only if N ^ 9(G). (c) The Frattini subgroup of a group is nilpotent. (d) If G has a normal subgroup N ^ (G) such that G/N is nilpotent, then G is itself nilpotent.
A note on p-nilpotence and solvability of finite groups
Journal of Algebra, 2009
In this note, we first give some examples to show that some hypotheses of some well-known results for a group G to be pnilpotent, solvable and supersolvable are essential and cannot be removed. Second, we give some generalizations of two theorems in [A. Ballester-Bolinches, X. Guo, Some results on p-nilpotence and solubility of finite groups, J. Algebra 228 (2000) 491-496].
Bulletin of the Australian Mathematical Society, 1969
In a finite soluble group G, the Fitting (or nilpotency) length h(G) can be considered as a measure for how strongly G deviates from being nilpotent. As another measure for this, the number v(G) of conjugacy classes of the maximal nilpotent subgroups of G may be taken. It is shown that there exists an integer-valued function f on the set of positive integers such that h(G) ≦ f(v(G)) for all finite (soluble) groups of odd order. Moreover, if all prime divisors of the order of G are greater than v(G)(v(G) - l)/2, then h(G) ≦3. The bound f(v(G)) is just of qualitative nature and by far not best possible. For v(G) = 2, h(G) = 3, some statements are made about the structure of G.
On the σ-Length of Maximal Subgroups of Finite σ-Soluble Groups
Mathematics
Let σ={σi:i∈I} be a partition of the set P of all prime numbers and let G be a finite group. We say that G is σ-primary if all the prime factors of |G| belong to the same member of σ. G is said to be σ-soluble if every chief factor of G is σ-primary, and G is σ-nilpotent if it is a direct product of σ-primary groups. It is known that G has a largest normal σ-nilpotent subgroup which is denoted by Fσ(G). Let n be a non-negative integer. The n-term of the σ-Fitting series of G is defined inductively by F0(G)=1, and Fn+1(G)/Fn(G)=Fσ(G/Fn(G)). If G is σ-soluble, there exists a smallest n such that Fn(G)=G. This number n is called the σ-nilpotent length of G and it is denoted by lσ(G). If F is a subgroup-closed saturated formation, we define the σ-F-lengthnσ(G,F) of G as the σ-nilpotent length of the F-residual GF of G. The main result of the paper shows that if A is a maximal subgroup of G and G is a σ-soluble, then nσ(A,F)=nσ(G,F)−i for some i∈{0,1,2}.
Nilpotency: A Characterization Of The Finite p-Groups
Journal of Mathematics , 2017
Abstract As parts of the characterizations of the finite p-groups is the fact that every finite p-group is NILPOTENT. Hence, there exists a derived series (Lower Central) which terminates at e after a finite number of steps. Suppose that G is a p-group of class at least m ≥ 3. Then L m-1G is abelian and hence G possesses a characteristic abelian subgroup which is not contained in Z(G). If L 3(G) = 1 such that pm is the highest order of an element of G/L2 (G) (where G is any p-group) then no element of L2(G) has an order higher than pm. [1]
On minimal non-p-closed groups and related properties
Publicationes Mathematicae Debrecen, 2011
Let p be a prime. A group is called p-closed if it has a normal Sylow p-subgroup and it is called p-exponent closed if the elements of order dividing p form a subgroup. A group is minimal non-p-closed if it is not p-closed but its proper subgroups and homomorphic images are. Similarly, a group is called minimal non-p-exponent closed if it is not p-exponent closed but all its proper subgroups and homomorphic images are. In this paper we characterize finite minimal non-p-closed groups and investigate the relationship between them and minimal non-p-exponent closed groups. In particular, we show that every minimal non-p-closed group is non-p-exponent closed and that minimal non-p-closed groups and simple minimal non-p-exponent closed groups have cyclic Sylow p-subgroups. Furthermore, given a prime p, we describe non-p-exponent closed groups of smallest order and we show that they coincide with non-p-closed groups of smallest order.
Some necessary and sufficient conditions for p-nilpotence of finite groups
Bulletin of the Australian Mathematical Society, 2003
The purpose of this paper is to give some necessary and sufficient conditions for p-nilpotent groups. We extend some results, including the well-known theorems of Burnside and Frobenius as well as some very recent theorems. We also apply our results to determine the structure of some finite groups in terms of formation theory.