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1.
Let G be a polycyclic group and α a regular automorphism of order four of G. If the map φ: G→ G defined by g~φ= [g, α] is surjective, then the second derived group of G is contained in the centre of G. Abandoning the condition on surjectivity, we prove that C_G(α~2) and G/[G, α~2] are both abelian-by-finite.  相似文献   

2.
n this paper, we study the structure of polycyclic groups admitting an automorphism of order four on the basis of Neumann’s result, and prove that if α is an automorphism of order four of a polycyclic group G and the map φ: GG defined by gφ = [g,α] is surjective, then G contains a characteristic subgroup H of finite index such that the second derived subgroup H″ is included in the centre of H and CH(α2) is abelian, both CG(α2) and G/[G, α2] are abelian-by-finite. These results extend recent and classical results in the literature.  相似文献   

3.
Let G be a group of finite generic rank, φ an injective endomorphism of the group G, and G(φ) the descending HNN-extension of G corresponding to the endomorphism φ. Let the index of the subgroup in G be finite and equal to n. It is proved that, if the group G is almost residually π-finite for some set π of primes coprime to n, then the group G(φ) is residually finite. This generalizes a series of known results, including the Wise-Hsu theorem on the residual finiteness of an arbitrary descending HNN-extension of any almost polycyclic group.  相似文献   

4.
A group is said to have finite (special) rank ≤ sif all of its finitely generated subgroups can be generated byselements. LetGbe a locally finite group and suppose thatH/HGhas finite rank for all subgroupsHofG, whereHGdenotes the normal core ofHinG. We prove that thenGhas an abelian normal subgroup whose quotient is of finite rank (Theorem 5). If, in addition, there is a finite numberrbounding all of the ranks ofH/HG, thenGhas an abelian subgroup whose quotient is of finite rank bounded in terms ofronly (Theorem 4). These results are based on analogous theorems on locally finitep-groups, in which case the groupGis also abelian-by-finite (Theorems 2 and 3).  相似文献   

5.
Let G be a finite group. The prime graph of G is denoted by Γ(G). The main result we prove is as follows: If G is a finite group such that Γ(G) = Γ(L 10(2)) then G/O 2(G) is isomorphic to L 10(2). In fact we obtain the first example of a finite group with the connected prime graph which is quasirecognizable by its prime graph. As a consequence of this result we can give a new proof for the fact that the simple group L 10(2) is uniquely determined by the set of its element orders.  相似文献   

6.
Edmonds showed that two free orientation preserving smooth actions φ1 and φ2 of a finite Abelian group G on a closed connected oriented smooth surface M are equivalent by an equivariant orientation preserving diffeomorphism iff they have the same bordism class [M,φ1]=[M,φ2] in the oriented bordism group Ω2(G) of the group G. In this paper, we compute the bordism class [M,φ] for any such action of G on M and we determine for a given M, the bordism classes in Ω2(G) that are representable by such actions of G on M. This will enable us to obtain a formula for the number of inequivalent such actions of G on M. We also determine the “weak” equivalence classes of such actions of G on M when all the p-Sylow subgroups of G are homocyclic (i.e. of the form n(Z/pαZ)).  相似文献   

7.
Let Mn be the algebra of all n×n matrices, and let φ:MnMn be a linear mapping. We say that φ is a multiplicative mapping at G if φ(ST)=φ(S)φ(T) for any S,TMn with ST=G. Fix GMn, we say that G is an all-multiplicative point if every multiplicative linear bijection φ at G with φ(In)=In is a multiplicative mapping in Mn, where In is the unit matrix in Mn. We mainly show in this paper the following two results: (1) If GMn with detG=0, then G is an all-multiplicative point in Mn; (2) If φ is an multiplicative mapping at In, then there exists an invertible matrix PMn such that either φ(S)=PSP-1 for any SMn or φ(T)=PTtrP-1 for any TMn.  相似文献   

8.
For a family of group words w we show that if G is a profinite group in which all w-values are contained in a union of finitely many subgroups with a prescribed property, then the verbal subgroup w(G) has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank. If G contains finitely many subgroups G 1, G 2, . . . , G s of finite exponent e whose union contains all γ k -values in G, it is shown that γ k (G) has finite (e, k, s)-bounded exponent. If G contains finitely many subgroups G 1, G 2, . . . , G s of finite rank r whose union contains all γ k -values, it is shown that γ k (G) has finite (k, r, s)-bounded rank.  相似文献   

9.
Let G be a finite group. The main result of this paper is as follows: If G is a finite group, such that Γ(G) = Γ(2G2(q)), where q = 32n+1 for some n ≥ 1, then G has a (unique) nonabelian composition factor isomorphic to 2 G 2(q). We infer that if G is a finite group satisfying |G| = |2 G 2(q)| and Γ(G) = Γ (2 G 2(q)) then G ? = 2 G 2(q). This enables us to give new proofs for some theorems; e.g., a conjecture of W. Shi and J. Bi. Some applications of this result are also considered to the problem of recognition by element orders of finite groups.  相似文献   

10.
We show that if G is a group of finite Morley rank, then the verbal subgroup <w(G)> is of finite width, where w is a concise word. As a byproduct, we show that if G is any abelian-by-finite group, then G n =<x n (G)> is definable. Received: 15 March 1996 / Published online: 18 July 2001  相似文献   

11.
If Λ is a ring and A is a Λ-module, then a terminal completion of Ext1Λ(A, ) is shown to exist if, and only if, ExtjΛ(A, P)=0 for all projective Λ-modules P and all sufficiently large j. Such a terminal completion exists for every A if, and only if, the supremum of the injective lengths of all projective Λ-modules, silp Λ, is finite. Analogous results hold for Ext1Λ(,A) and involve spli Λ, the supremum of the projective lengths of the injective Λ-modules. When Λ is an integral group ring ZG, spliZG is finite implies silp ZG is finite. Also the finiteness of spli is preserved under group extensions. If G is a countable soluble group, the spli ZG is finite if, and only if, the Hirsch number of G is finite.  相似文献   

12.
We generalize the fundamental theorem for Burnside rings to the mark morphism of plus constructions defined by Boltje. The main observation is the following: If D is a restriction functor for a finite group G, then the mark morphism φ:D+D+ is the same as the norm map of the Tate cohomology sequence (over conjugation algebra for G) after composing with a suitable isomorphism of D+. As a consequence, we obtain an exact sequence of Mackey functors
  相似文献   

13.
We describe the structure of the group U n of unitriangular automorphisms of the relatively free group G n of finite rank n in an arbitrary variety C of groups. This enables us to introduce an effective concept of normal form for the elements and present U n by using generators and defining relations. The cases n = 1, 2 are obvious: U 1 is trivial, and U 2 is cyclic. For n ?? 3 we prove the following: If G n?1 is a nilpotent group then so is U n . If G n?1 is a nilpotent-by-finite group then U n admits a faithful matrix representation. But if the variety C is different from the variety of all groups and G n?1 is not nilpotent-by-finite then U n admits no faithful matrix representation over any field. Thus, we exhaustively classify linearity for the groups of unitriangular automorphisms of finite rank relatively free groups in proper varieties of groups, which complements the results of Olshanskii on the linearity of the full automorphism groups AutG n . Moreover, we introduce the concept of length of an automorphism of an arbitrary relatively free group G n and estimate the length of the inverse automorphism in the case that it is unitriangular.  相似文献   

14.
Let R be a commutative ring, M an R-module and G a group of R-automorphisms of M, usually with some sort of rank restriction on G. We study the transfer of hypotheses between M/C M (G) and [M,G] such as Noetherian or having finite composition length. In this we extend recent work of Dixon, Kurdachenko and Otal and of Kurdachenko, Subbotin and Chupordia. For example, suppose [M,G] is R-Noetherian. If G has finite rank, then M/C M (G) also is R-Noetherian. Further, if [M,G] is R-Noetherian and if only certain abelian sections of G have finite rank, then G has finite rank and is soluble-by-finite. If M/C M (G) is R-Noetherian and G has finite rank, then [M,G] need not be R-Noetherian.  相似文献   

15.
The following result is proved: If either G is a finite abelian group or a semidirect product of a cyclic group of prime order by a finite abelian group of odd order, then every connected Cayley graph of G is hamiltonian.  相似文献   

16.
Let G be a finite group and π e (G) be the set of element orders of G. Let k ∈ π e (G) and m k be the number of elements of order k in G. Set nse(G):= {m k : k ∈ π e (G)}. In fact nse(G) is the set of sizes of elements with the same order in G. In this paper, by nse(G) and order, we give a new characterization of finite projective special linear groups L 2(p) over a field with p elements, where p is prime. We prove the following theorem: If G is a group such that |G| = |L 2(p)| and nse(G) consists of 1, p 2 ? 1, p(p + ?)/2 and some numbers divisible by 2p, where p is a prime greater than 3 with p ≡ 1 modulo 4, then G ? L 2(p).  相似文献   

17.
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   

18.
We prove that if (H, G) is a small, nm-stable compact G-group, then H is nilpotent-by-finite, and if additionally NM(H) < ω or NM(H) = ω α for some ordinal α, then H is abelian-by-finite. Both results are significant steps towards the proof of the conjecture that each small, nm-stable compact G-group is abelian-by-finite. We provide counter-examples to the NM-gap conjecture, that is we give examples of small, nm-stable compact G-groups of infinite ordinal NM-rank.  相似文献   

19.
Fucai Lin 《Semigroup Forum》2014,88(1):273-278
A topological space G is said to be a rectifiable space provided that there are a surjective homeomorphism φ:G×GG×G and an element eG such that π 1°φ=π 1 and for every xG we have φ(x,x)=(x,e), where π 1:G×GG is the projection to the first coordinate. Let G be a rectifiable space and C(G) be the family of all non-empty compact subsets of G. In this paper, we study the Vietoris topology on C(G), and show that if G is a locally compact rectifiable space, then (C(G),?) together with the Vietoris topology is a topological semi-right loop.  相似文献   

20.
For any finite group G, we construct a finite poset (or equivalently, a finite T0-space) X, whose group of automorphisms is isomorphic to G. If the order of the group is n and it has r generators, X has n(r+2) points. This construction improves previous results by G. Birkhoff and M.C. Thornton. The relationship between automorphisms and homotopy types is also analyzed.  相似文献   

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