共查询到20条相似文献,搜索用时 15 毫秒
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.
Mathematische Zeitschrift - 相似文献
3.
4.
S. V. Larin 《Mathematical Notes》1977,21(2):132-137
For a wide class of Abelian groups, necessary and sufficient conditions under which a group admits an automorphism of order
qn are found; we also present necessary and sufficient conditions under which a group admits an automorphism ϕ of order qn such that ϕqm is a fixed-point-free automorphism for some m < n.
Translated from Matematicheskie Zametki, Vol. 21, No. 2, pp. 239–249, February, 1977. 相似文献
5.
6.
7.
8.
9.
Suppose that a finite group G admits a Frobenius group of automorphisms BC of coprime order with kernel B and complement C such that C G (C) is abelian. It is proved that if B is abelian of rank at least two and \({[C_G(u), C_G(v),\dots,C_G(v)]=1}\) for any \({u,v\in B{\setminus}\{1\}}\), where C G (v) is repeated k times, then G is nilpotent of class bounded in terms of k and |C| only. It is also proved that if B is abelian of rank at least three and C G (b) is nilpotent of class at most c for every \({b \in B{\setminus}\{1\}}\), then G is nilpotent of class bounded in terms of c and |C|. The proofs are based on results on graded Lie rings with many commuting components. 相似文献
10.
P. Shumyatsky’s question 11.126 in the “Kourovka Notebook” is answered in the affirmative: it is proved that there exist a
constant c and a function of a positive integer argument f(m) such that if a finite group G admits an automorphism ϕ of order
4 having exactly m fixed points, then G has a normal series G ⩾ H ⩽ N such that |G/H| ⩽ f(m), the quotient group H/N is nilpotent
of class ⩽ 2, and the subgroup N is nilpotent of class ⩽ c (Thm. 1). As a corollary we show that if a locally finite group
G contains an element of order 4 with finite centralizer of order m, then G has the same kind of a series as in Theorem 1.
Theorem 1 generalizes Kovács’ theorem on locally finite groups with a regular automorphism of order 4, whereby such groups
are center-by-metabelian. Earlier, the first author proved that a finite 2-group with an almost regular automorphism of order
4 is almost center-by-metabelian. The proof of Theorem 1 is based on the authors’ previous works dealing in Lie rings with
an almost regular automorphism of order 4. Reduction to nilpotent groups is carried out by using Hall-Higman type theorems.
The proof also uses Theorem 2, which is of independent interest, stating that if a finite group S contains a nilpotent subgroup
T of class c and index |S: T | = n, then S contains also a characteristic nilpotent subgroup of class ⩽ c whose index is bounded
in terms of n and c. Previously, such an assertion has been known for Abelian subgroups, that is, for c = 1.
__________
Translated from Algebra i Logika, Vol. 45, No. 5, pp. 575–602, September–October, 2006. 相似文献
11.
V. S. Atabekyan 《Mathematical Notes》2014,95(5-6):586-589
It is proved that, if the order of a splitting automorphism of odd period n ≥ 1003 of a free Burnside group B(m, n) is equal to a power of some prime, then the automorphism is inner. Thus, an affirmative answer is given to the question concerning the coincidence of the splitting automorphisms of the group B(m, n) with the inner automorphisms for all automorphisms of order p k (p is a prime). This question was posed in 1990 in “Kourovka Notebook” (see the 11th edition, Question 11.36.b). 相似文献
12.
Suppose that a finite group G admits a Frobenius group of automorphisms BC of coprime order with kernel B and complement C such that C G (C) is abelian. It is proved that if B is abelian of rank at least two and [CG(u), CG(v),...,CG(v)]=1{[C_G(u), C_G(v),dots,C_G(v)]=1} for any u,v ? B{1}{u,vin B{setminus}{1}}, where C G (v) is repeated k times, then G is nilpotent of class bounded in terms of k and |C| only. It is also proved that if B is abelian of rank at least three and C G (b) is nilpotent of class at most c for every b ? B{1}{b in B{setminus}{1}}, then G is nilpotent of class bounded in terms of c and |C|. The proofs are based on results on graded Lie rings with many commuting components. 相似文献
13.
I. Kh. Bekker 《Mathematical Notes》1997,61(4):399-408
In the paper, sufficient conditions for the splittability of mixed Abelian groups with periodic automorphism groups are established.
Classes of mixed splittable Abelian groups with perfect holomorphs are distinguished.
Translated fromMaternaticheskie Zametki, Vol. 61, No. 4, pp. 483–493, April, 1997.
Translated by A. I. Shtern 相似文献
14.
Vladimir D Tonchev 《Journal of Combinatorial Theory, Series A》1985,40(1):62-81
The only primes which can divide the order of the automorphism group of a Hadamard matrix of order 28 are 13, 7, 3, and 2, and there are precisely four inequivalent matrices with automorphisms of order 13 (Tonchev, J. Combin. Theory Ser. A35 (1983), 43–57). In this paper we show that there are exactly twelve inequivalent Hadamard matrices of order 28 with automorphisms of order 7. In particular, there are precisely seven matrices with transitive automorphism groups. 相似文献
15.
Letk be a locally compact, totally disconnected, nondiscrete field, and letG be a Lie group overk satisfying suitable conditions which depend on the characteristic ofk. It is shown thatG is compact if it admits a bicontinuous automorphism which is ergodic with respect to Haar measure.
Research supported in part by NSF Grant GP 20150 for the second-named author. 相似文献
16.
We introduce the so-called multiscale limit for spectral curves associated with real finite-gap sine-Gordon solutions. This
technique allows us to solve the old problem of calculating the density of the topological charge for real finite-gap sine-Gordon
solutions directly from the θ-functional formulae. 相似文献
17.
18.
Alexander Berenstein 《Archive for Mathematical Logic》2007,46(3-4):289-299
Let G be a countable group. We prove that there is a model companion for the theory of Hilbert spaces with a group G of automorphisms. We use a theorem of Hulanicki to show that G is amenable if and only if the structure induced by countable copies of the regular representation of G is existentially closed. 相似文献
19.
20.
D. G. Khramtsov 《Mathematical Notes》1985,38(3):721-724