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1.
Let G be a finite soluble group and P a subgroup of order 3. In this article we prove some results about the soluble groups generated by 2 conjugates of P and we use these results to produce some properties of G.  相似文献   

2.
Let G be a fnite group and P a subgroup of order 3. In this paper we proved some results about the soluble subgroups generated by three conjugates of P and we use these results to produce some properties of the group G.  相似文献   

3.
《代数通讯》2013,41(12):5795-5798
We conjecture that a finitely generated relatively free group G has a finitely generated commutator subgroup G′ if and only if G satisfies a positive law. We confirm this conjecture for groups G in the large class, containing all residually finite and all soluble groups.  相似文献   

4.
《代数通讯》2013,41(7):3287-3293
Abstract

For an element a of a group G,let S(a) denote the semigroup generated by all conjugates of a in G. We prove that if G is solvable of finite rank and 1 ? S(a) for all 1 ≠ a ∈ G,then ?a G ?/?b G ? is a periodic group for every b ∈ S(a). Conversely if every two generator subgroup of a finitely generated torsion-free solvable group G has this property then G has finite rank,and if every finitely generated subgroup has this property then every partial order on G can be extended to a total order.  相似文献   

5.
In this note we show that if Gis a finitely generated soluble group, then every infinite subset of Gcontains two elements generating a nilpotent group of class at most kif and only if Gis finite by a group in which every two generator subgroup is nilpotent of class at most k.  相似文献   

6.
7.
A subgroup X of a group G is called pronormal-by-finite if there exists a pronormal subgroup Y of G such that YX and |X : Y| is finite. The structure of (generalized) soluble groups in which all subgroups are pronormal-by-finite is investigated. Among other results, it is proved in particular that a finitely generated soluble group with such property is central-by-finite, provided that it has no infinite dihedral sections.  相似文献   

8.
Let G be a finite group and H a subgroup of G. We say that: (1) H is τ-quasinormal in G if H permutes with all Sylow subgroups Q of G such that (|Q|, |H|) = 1 and (|H|, |Q G |) ≠ 1; (2) H is weakly τ-quasinormal in G if G has a subnormal subgroup T such that HT = G and THH τG , where H τG is the subgroup generated by all those subgroups of H which are τ-quasinormal in G. Our main result here is the following. Let ℱ be a saturated formation containing all supersoluble groups and let XE be normal subgroups of a group G such that G/E ∈ ℱ. Suppose that every non-cyclic Sylow subgroup P of X has a subgroup D such that 1 < |D| < |P| and every subgroup H of P with order |H| = |D| and every cyclic subgroup of P with order 4 (if |D| = 2 and P is non-Abelian) not having a supersoluble supplement in G is weakly τ-quasinormal in G. If X is either E or F* (E), then G ∈ ℱ.  相似文献   

9.
Let ϕ be an automorphism of prime order p of the group G with C G (ϕ) finite of order n. We prove the following. If G is soluble of finite rank, then G has a nilpotent characteristic subgroup of finite index and class bounded in terms of p only. If G is a group with finite Hirsch number h, then G has a soluble characteristic subgroup of finite index in G with derived length bounded in terms of p and n only and a soluble characteristic subgroup of finite index in G whose index and derived length are bounded in terms of p, n and h only. Here a group has finite Hirsch number if it is poly (cyclic or locally finite). This is a stronger notion than that used in [Wehrfritz B.A.F., Almost fixed-point-free automorphisms of order 2, Rend. Circ. Mat. Palermo (in press)], where the case p = 2 is discussed.  相似文献   

10.
In [2] we proved that ifG is a finite group containing an involution whose centralizer has order bounded by some numberm, thenG contains a nilpotent subgroup of class at most two and index bounded in terms ofm. One of the steps in the proof of that result was to show that ifG is soluble, then ¦G/F(G) ¦ is bounded by a function ofm, where F (G) is the Fitting subgroup ofG. We now show that, in this part of the argument, the involution can be replaced by an arbitrary element of prime order.  相似文献   

11.
Let G be a finitely generated group, and A a ?[G]-module of flat dimension n such that the homological invariant Σ n (G, A) is not empty. We show that A has projective dimension n as a ?[G]-module. In particular, if G is a group of homological dimension hd(G) = n such that the homological invariant Σ n (G, ?) is not empty, then G has cohomological dimension cd(G) = n. We show that if G is a finitely generated soluble group, the converse is true subject to taking a subgroup of finite index, i.e., the equality cd (G) = hd(G) implies that there is a subgroup H of finite index in G such that Σ(H, ?) ≠ ?.  相似文献   

12.
We call a subgroup H of a finite group G c-supplemented in G if there exists a subgroup K of G such that G = HK and HK ⩽ core(H). In this paper it is proved that a finite group G is p-nilpotent if G is S 4-free and every minimal subgroup of PG N is c-supplemented in N G (P), and when p = 2 P is quaternion-free, where p is the smallest prime number dividing the order of G, P a Sylow p-subgroup of G. As some applications of this result, some known results are generalized.  相似文献   

13.
Let \(X\) be an infinite set, \(f\) a partial one-to-one transformation of \(X\), and \(H\) a normal subgroup of G X , the group of all permutations of \(X\). We investigate when \(H\) is equal to \(G_{<f:H>}\). That is, we are interested when \(H\) is the full group of normalizers of the semigroup of transformations on \(X\) generated by conjugates of \(f\) by elements of \(H\).  相似文献   

14.
LetG be a finite 2-group. If each normal abelian subgroup ofG can be generated byd elements, then each subgroup ofG can be generated byd 2+1/2d(d+1) elements.  相似文献   

15.
《代数通讯》2013,41(12):4769-4784
Abstract

Neumann characterized the groups in which every subgroup has finitely many conjugates only as central-by-finite groups. If 𝔛 is a class of groups, a group G is said to have 𝔛-conjugate classes of subgroups if G/Core G (N G (H)) ∈ 𝔛 for every subgroup H of G. In this paper, we generalize Neumann's result by showing that a group has polycyclic-by-finite classes of conjugate subgroup if and only if it is central-by-(polycyclic-by-finite).  相似文献   

16.
Let ? be a class of groups and G a finite group. We call a set Σ of subgroups of G a G-covering subgroup system for ? if G ∈ ? whenever Σ ? ?. For a non-identity subgroup H of G, we put Σ H be some set of subgroups of G which contains at least one supplement in G of each maximal subgroup of H. Let p ≠ q be primes dividing |G|, P, and Q be non-identity a p-subgroup and a q-subgroup of G, respectively. We prove that Σ P and Σ P  ∪ Σ Q are G-covering subgroup systems for many classes of finite groups.  相似文献   

17.
I. Levi  R.B. McFadden 《代数通讯》2013,41(10):4829-4838
It is well known that the symmetric group S ntogether with one idempotent of rank n- 1 on a finite n-element set Nserves as a set of generators for the semigroup T nof all the total transformations on N. It is also well known that the singular part Sing n of T n can be generated by a set of idempotents of rank n- 1. The purpose of this paper is to begin an investigation of the way in which Singnand its subsemigroups can be generated by the conjugates of a subset of elements of T n by a subgroup of S n . We look for the smallest subset of elements of T n that will serve and, correspondingly, for a characterization of those subgroups of S n that will serve. Using some techniques from graph theory we prove our main result:the conjugates of a single transformation of rank n- 1 under Gsuffice to generate Singnif and only if Gis what we define to be a 2-block transitive subgroup of S n .  相似文献   

18.
Gil Kaplan  Dan Levy 《代数通讯》2013,41(5):1517-1523
We prove conditions for a product of distinct subgroups of an arbitrary group G to be a subgroup of G. In particular, the normal closure of any A ≤ G is equal to the product of some distinct conjugates of A. As an application of the later result we derive constraints on the size of a nontrivial conjugacy class of a finite non-Abelian simple group.  相似文献   

19.
A finite group G is said to be a PST-group if every subnormal subgroup of G permutes with every Sylow subgroup of G. We shall discuss the normal structure of soluble PST-groups, mainly defining a local version of this concept. A deep study of the local structure turns out to be crucial for obtaining information about the global property. Moreover, a new approach to soluble PT-groups, i.e., soluble groups in which permutability is a transitive relation, follows naturally from our vision of PST-groups. Our techniques and results provide a unified point of view for T-groups, PT-groups, and PST-groups in the soluble universe, showing that the difference between these classes is quite simply their Sylow structure.  相似文献   

20.
Let G be a finite group. We fix in every noncyclic Sylow subgroup P of G some subgroup D satisfying 1 < |D| < |P| and study the structure of G under the assumption that all subgroups H of P with |H| = |D| are c-normal in G.  相似文献   

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