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1.
2.
M. V. Neshadim 《Algebra and Logic》1996,35(5):316-318
An automorphism of an arbitrary group is called normal if all subgroups of this group are left invariant by it. Lubotski [1]
and Lue [2] showed that every normal automorphism of a noncyclic free group is inner. Here we prove that every normal automorphism
of a nontrivial free product of groups is inner as well.
Supported by RFFR grant No. 13-011-1513.
Translated fromAlgebra i Logika, Vol. 35, No. 5, pp. 562–566, September–October, 1996. 相似文献
3.
4.
Vladimir Tolstykh 《Journal of Pure and Applied Algebra》2011,215(9):2086-2098
Let F be an infinitely generated free group and let R be a fully invariant subgroup of F such that (a) R is contained in the commutator subgroup F′ of F and (b) the quotient group F/R is residually torsion-free nilpotent. Then the automorphism group of the group F/R′ is complete. In particular, the automorphism group of any infinitely generated free solvable group of derived length at least two is complete.This extends a result by Dyer and Formanek (1977) [7] on finitely generated groups Fn/R′ where Fn is a free group of finite rank n at least two and R a characteristic subgroup of Fn. 相似文献
5.
Let F be a free Lie algebra of rank n ≥ 2 and A be a free abelian Lie algebra of rank m ≥ 2. We prove that the test rank of the abelian product F ×A is m. Morever we compute the test rank of the algebra F/gk( F) ¢F/\gamma _{k}\left( F\right) ^{^{\prime }}. 相似文献
6.
《代数通讯》2013,41(9):2957-2975
ABSTRACT Let F m (N) be the free left nilpotent (of class two) Leibniz algebra of finite rank m, with m ≥ 2. We show that F m (N) has non-tame automorphisms and, for m ≥ 3, the automorphism group of F m (N) is generated by the tame automorphisms and one more non-tame IA-automorphism. Let F(N) be the free left nilpotent Leibniz algebra of rank greater than 1 and let G be an arbitrary non-trivial finite subgroup of the automorphism group of F(N). We prove that the fixed point subalgebra F(N) G is not finitely generated. 相似文献
7.
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
8.
E. I. Khukhro 《Siberian Mathematical Journal》2002,43(5):955-962
We prove that if a finite group G of rank r admits an automorphism of prime order having exactly m fixed points, then G has a -invariant subgroup of (r,m)-bounded index which is nilpotent of r-bounded class (Theorem 1). Thus, for automorphisms of prime order the previous results of Shalev, Khukhro, and Jaikin-Zapirain are strengthened. The proof rests, in particular, on a result about regular automorphisms of Lie rings (Theorem 3). The general case reduces modulo available results to the case of finite p-groups. For reduction to Lie rings powerful p-groups are also used. For them a useful fact is proved which allows us to glue together nilpotency classes of factors of certain normal series (Theorem 2). 相似文献
9.
V. S. Atabekyan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2010,45(2):112-122
In the present paper for arbitrary automorphism φ of the free Bunside group B(m, n) and for any odd number n ≥ 1003 a sufficient condition for existence of non-φ-admissible normal subgroup of B(m, n) was found. In particular, if automorphism φ is normal, then for any basis {a
1, a
2, …, a
m
} of the group B(m, n) there is an integer k such that for each i the elements a
i
and φ(a
i)
k
are conjugates. 相似文献
10.
In Theorem 1, letting p be a prime, we prove: (1) If G=Sn is a symmetric group of degree n, then G contains two Sylow p-subgroups with trivial intersection iff (p, n) ∉ {(3, 3), (2,
2), (2, 4), (2, 8)}, and (2) If H=An is an alternating group of degree n, then H contains two Sylow p-subgroups with trivial intersection iff (p, n) ∉ {(3, 3),
(2, 4)}. In Theorem 2, we argue that if G is a finite simple non-Abelian group and p is a prime, then G contains a pair of
Sylow p-subgroups with trivial intersection. Also we present the corollary which says that if P is a Sylow subgroup of a finite
simple non-Abelian group G, then ‖G‖>‖P‖2.
Supported by RFFR grants Nos. 93-01-01529, 93-01-01501, and 96-01-01893, and by International Science Foundation and Government
of Russia grant RPC300.
Translated fromAlgebra i Logika, Vol. 35, No. 4, pp. 424–432, July–August, 1996. 相似文献
11.
V. D. Mazurov 《Algebra and Logic》2006,45(2):117-123
Let G be a group. A subset X of G is called an A-subset if X consists of elements of order 3, X is invariant in G, and every
two non-commuting members of X generate a subgroup isomorphic to A4 or to A5. Let X be the A-subset of G. Define a non-oriented graph Γ(X) with vertex set X in which two vertices are adjacent iff they
generate a subgroup isomorphic to A4. Theorem 1 states the following. Let X be a non-empty A-subset of G. (1) Suppose that C is a connected component of Γ(X)
and H = 〈C〉. If H ∩ X does not contain a pair of elements generating a subgroup isomorphic to A5 then H contains a normal elementary Abelian 2-subgroup of index 3 and a subgroup of order 3 which coincides with its centralizer
in H. In the opposite case, H is isomorphic to the alternating group A(I) for some (possibly infinite) set I, |I| ≥ 5. (2)
The subgroup 〈XG〉 is a direct product of subgroups 〈C
α〉-generated by some connected components C
α of Γ(X). Theorem 2 asserts the following. Let G be a group and X⊆G be a non-empty G-invariant set of elements of order 5 such that every two non-commuting members of X generate a subgroup
isomorphic to A5. Then 〈XG〉 is a direct product of groups each of which either is isomorphic to A5 or is cyclic of order 5.
Supported by RFBR grant No. 05-01-00797; FP “Universities of Russia,” grant No. UR.04.01.028; RF Ministry of Education Developmental
Program for Scientific Potential of the Higher School of Learning, project No. 511; Council for Grants (under RF President)
and State Aid of Fundamental Science Schools, project NSh-2069.2003.1.
__________
Translated from Algebra i Logika, Vol. 45, No. 2, pp. 203–214, March–April, 2006. 相似文献
12.
We consider locally nilpotent periodic groups admitting an almost regular automorphism of order 4. The following are results
are proved: (1) If a locally nilpotent periodic group G admits an automorphism ϕ of order 4 having exactly m<∞ fixed points,
then (a) the subgroup {ie176-1} contains a subgroup of m-bounded index in {ie176-2} which is nilpotent of m-bounded class,
and (b) the group G contains a subgroup V of m-bounded index such that the subgroup {ie176-3} is nilpotent of m-bounded class
(Theorem 1); (2) If a locally nilpotent periodic group G admits an automorphism ϕ of order 4 having exactly m<∞ fixed points,
then it contains a subgroup V of m-bounded index such that, for some m-bounded number f(m), the subgroup {ie176-4}, generated
by all f(m) th powers of elements in {ie176-5} is nilpotent of class ≤3 (Theorem 2).
Supported by RFFR grant No. 94-01-00048 and by ISF grant NQ7000.
Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 314–333, May–June, 1996. 相似文献
13.
Theorem A:If ℬ is an infinite Moufang polygon of finite Morley rank, then ℬ is either the projective plane, the symplectic quadrangle,
or the split Cayley hexagon over some algebraically closed field. In particular, ℬ is an algebraic polygon.
It follows that any infinite simple group of finite Morley rank with a spherical MoufangBN-pair of Tits rank 2 is eitherPSL
3(K),PSp
4(K) orG
2(K) for some algebraically closed fieldK.
Spherical irreducible buildings of Tits rank ≥ 3 are uniquely determined by their rank 2 residues (i.e. polygons). Using Theorem
A we show
Theorem B:If G is an infinite simple group of finite Morley rank with a spherical Moufang BN-pair of Tits rank ≥ 2, then G is (interpretably)
isomorphic to a simple algebraic group over an algebraically closed field.
Theorem C:Let K be an infinite field, and let G(K) denote the group of K-rational points of an isotropic adjoint absolutely simple K-algebraic
group G of K-rank ≥ 2. Then G(K) has finite Morley rank if and only if the field K is algebraically closed.
We also obtain a result aboutBN-pairs in splitK-algebraic groups: such aBN-pair always contains the root groups. Furthermore, we give a proof that the sets of points, lines and flags of any ℵ1-categorical polygon have Morley degree 1.
Partially sponsored by the Edmund Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation
(Germany).
Supported by the Minerva Foundation (Germany).
Research Director at the Fund for Scientific Research-Flanders (Belgium). 相似文献
14.
E. P. Vdovin 《Algebra and Logic》1999,38(2):67-83
We bring out upper bounds for the orders of Abelian subgroups in finite simple groups. (For alternating and classical groups,
these estimates are, or are nearly, exact.) Precisely, the following result, Theorem A, is proved. Let G be a non-Abelian
finite simple group and G
L2 (q), where q=pt for some prime number p. Suppose A is an Abelian subgroup of G. Then |A|3<|G|. Our proof is based on a classification of finite simple groups. As a consequence we obtain Theorem B, which states that
a non-Abelian finite simple group G can be represented as ABA, where A and B are its Abelian subgroups, iff G≌ L2(2t) for some t ≥ 2; moreover, |A|-2t+1, |B|=2t, and A is cyclic and B an elementary 2-group.
Translated fromAlgebra i Logika, Vol. 38, No. 2, pp. 131–160, March–April, 1999. 相似文献
15.
John P. D’Angelo 《Journal of Geometric Analysis》2004,14(2):215-229
We first prove a uniqueness result for certain group-invariant CR mappings to hyperquadrics. For cyclic groups these mappings
lead to a collection of polynomials ƒp,q (with integer coefficients) in two variables; here p and q are positive integers. We use the uniqueness result to prove some
surprising number-theoretic results about the ƒp,q, in particular, ƒp,q is congruent to xP + yP modulo (p) (for P ≥ 2) if and only if p is prime. We also determine recurrence relations for these polynomials for q ≤ 3
and determine a functional equation for a generating function. Finally, we discuss the invariant polynomials that arise for
certain representations of dihedral groups to illustrate the non-Abelian case. 相似文献
16.
V. G. Bardakov 《Algebra and Logic》2000,39(4):224-251
We study commutator length in free groups. (By a commutator lengthcl(g) of an element g in a derived subgroup G′ of a group G we mean the least natural number k such that g is a product of k
commutators.) A purely algebraic algorithm is constructed for computing commutator length in a free group F2 (Thm. 1). Moreover, for every element z ε F′2 and for any natural m, the following estimate derives:cl(zm) ≥ (ms(z) + 6)/12, where s(z) is a nonnegative number defined by an element z (Thm. 2). This estimate is used to compute
commutator length of some particular elements. By analogy with the concept of width of a derived subgroup known in group theory,
we define the concept of width of a derived subalgebra. The width of a derived subalgebra is computed for an algebra P of
pairs, and also for its corresponding Lie algebra PL. The algebra of pairs arises naturally in proving Theorem 2 and enjoys a number of interesting properties. We state that
in a free group F2k with free generators a1, b1, ..., ak, bk, k εN, every natural m satisfiescl(([a1, b1] ... [ak, bk])m)=[(2 − m)/2] + mk. For k=1, this entails a known result of Culler. The notion of a growth function as applied to a finitely
generated group G is well known. Associated with a derived subgroup of F2 is some series depending on two variables which bears information not only on the number of elements of prescribed length
but also on the number of elements of prescribed commutator length. A number of open questions are formulated.
Supported by RFFR grant No. 98-01-00699.
Translated fromAlgebra i Logika, Vol. 39, No. 4, pp. 395–440, July–August, 2000. 相似文献
17.
For a prime number p let G be a profinite p-PD
n
group with a closed normal subgroup N such that G/N is a profinite p-PD
m
group and that H
i
(V, $
\mathbb{F}
$
\mathbb{F}
p
) is finite for every open subgroup V of N and all i ≤ [n/2]. Generalising [12, Thm. 3.7.4] we show that m ≤ n and N is a profinite p-PD
n − m
group.
In case that G is a pro-p
PD
n
group of Euler characteristic 0 with a closed normal subgroup N of type FP
[n−1 / 2] such that G/N is soluble-by-finite pro-p group of finite rank, we show that N is a pro-p
PD
n − m
group, where m = vcd
p
(G/N). As a corollary we obtain that a pro-p
PD
3 group with infinite abelianization is either soluble or contains a free nonprocyclic pro-p subgroup. 相似文献
18.
从有限Abel p-群P的型不变量出发,给出了其自同构群AutP的阶的计算公式,并利用|AutP|的计算公式得到了下面3个结果:1.由有限Abel p-群的型不变量的两种变换得到了其自同构群的阶的变化规律;2.用群的阶、秩、幂指数三个量界定了有限Abel p-群的自同构的阶;3.对部分Frattini子群为p阶群的有限p-群,确定了其自同构群的阶何时达到最小值和最大值. 相似文献
19.
Let F
n
be the free group of rank n, let Aut(F
n
) be its automorphism group and let Out(F
n
) be its outer automorphism group. We show that every solvable subgroup of Out(F
n
) has a finite index subgroup that is finitely generated and free Abelian. We also show that every Abelian subgroup of Out(F
n
) has a finite index subgroup that lifts to Aut(F
n
). 相似文献
20.
E. I. Bunina 《Journal of Mathematical Sciences》2010,169(5):589-613
In this paper, we prove that every automorphism of a Chevalley group of type B
l
, l ≥ 2, over a commutative local ring with 1/2 is standard, i.e., it is a composition of ring, inner, and central automorphisms. 相似文献