Abelian Subgroups of Garside Groups

In this article, we show that for every abelian subgroup H of a Garside group, some conjugate g ^?1 Hg consists of ultra summit elements and the centralizer of H is a finite index subgroup of the normalizer of H. Combining with the results on translation numbers in Garside groups, we obtain an easy proof of the algebraic flat torus theorem for Garside groups and solve several algorithmic problems concerning abelian subgroups of Garside groups.     

Commutator Invariant Subgroups of Abelian Groups

We describe the commutator invariant subgroups of a nonreduced abelian group. We find out when all commutator invariant subgroups of a separable group and an algebraically compact torsion-free group are fully invariant and describe the E-centers and E-commutants of these and some other groups.     

Recognizing Abelian Sylow Subgroups in Finite Groups

Let p be a prime. We prove that if a finite group G has non-abelian Sylow p-subgroups, and the class size of every p-element in G is coprime to p, then G contains a simple group as a subquotient which exhibits the same property. In addition, we provide a list of all the simple groups and primes such that the Sylow p-subgroups are non-abelian and all p-elements have class size coprime to p.     

On Primary Abelian Groups Modulo Finite Subgroups

We study the existence of several classes 𝒦 of Abelian p-groups, p a fixed prime, which possess the following property: A ∈ 𝒦?A/F ∈ 𝒦, whenever F is a finite subgroup of the Abelian p-group A.     

Large Abelian Unipotent Subgroups of Finite Chevalley Groups

We compute maximal orders of unipotent Abelian subgroups, estimate p-ranks, and describe the structure of Thompson subgroups of maximal unipotent subgroups of finite exceptional groups of Lie type.     

Classes of Subgroups of Simply Presented Primary Abelian Groups

A class 𝒳 of abelian p-groups is closed under ω₁-bijective homomorphisms if whenever f: G → H is a homomorphism with countable kernel and cokernel, then G ∈ 𝒳 iff H ∈ 𝒳. For an ordinal α, we consider the smallest class with this property containing (a) the p ^α-bounded simply presented groups; (b) the p ^α-projective groups; (c) the subgroups of p ^α-bounded simply presented groups. This builds upon classical results of Nunke from [14 Nunke , R. ( 1963 ). Purity and subfunctors of the identity . In: Topics in Abelian Groups , Chicago : Scott, Foresman and Co. , pp. 121 – 171 . [Google Scholar]] and [15 Nunke , R. ( 1967 ). Homology and direct sums of countable abelian groups . Math. Z. 101 : 182 – 212 .[Crossref], [Web of Science ®] , [Google Scholar]]. Particular attention is paid to the separable groups in these classes.     

Abelian Subgroups of Finitely Generated Kleinian Groups are Separable

By a Kleinian group we mean a discrete subgroup of PSL(2, C).We prove that abelian subgroups of finitely generated Kleiniangroups are separable. In other words, if M = H³/ is a hyperbolic3-orbifold, with finitely generated, then abelian subgroupsof are separable in . 1991 Mathematics Subject Classification20E26, 51M10, 57M05.     

交换子群与群的结构研究

交换子群的中心化子和正规化子对有限群的结构有非常重要的影响,给出若干由交换子群的中心化子或正规化子满足某些条件所确定的有限群的结构描述.     

Maximal Orders of Abelian Subgroups in Finite Chevalley Groups

In the present paper, for any finite group G of Lie type (except for ² F ₄(q)), the order a(G) of its large Abelian subgroup is either found or estimated from above and from below (the latter is done for the groups F ₄ (q), E ₆ (q), E ₇ (q), E ₈ (q), and ² E ₆(q ²)). In the groups for which the number a(G) has been found exactly, any large Abelian subgroup coincides with a large unipotent or a large semisimple Abelian subgroup. For the groups F ₄ (q), E ₆ (q), E ₇ (q), E ₈ (q), and ² E ₆(q ²)), it is shown that if an Abelian subgroup contains a noncentral semisimple element, then its order is less than the order of an Abelian unipotent group. Hence in these groups the large Abelian subgroups are unipotent, and in order to find the value of a(G) for them, it is necessary to find the orders of the large unipotent Abelian subgroups. Thus it is proved that in a finite group of Lie type (except for ² F ₄(q))) any large Abelian subgroup is either a large unipotent or a large semisimple Abelian subgroup.     

Groups Covered by an Infinite Number of Abelian Subgroups

If a group G is covered by Abelian subgroups ( an infinite cardinal) then and this estimate is sharp. This answers a question of Faber, Laver and McKenzie. Received February 1, 2000     

Direct Products with Abelian Automorphism Groups

The article considers when the direct product of two finite groups has an Abelian automorphism group.     

Symmetric Groups, Wreath Products, Morita Equivalences, and Broue's Abelian Defect Group Conjecture

It is shown that for any prime p, and any non-negative integerw less than p, there exist p-blocks of symmetric groups of defectw, which are Morita equivalent to the principal p-block of thegroup S_p S_w. Combined with work of J. Rickard, this provesthat Broué's abelian defect group conjecture holds forp-blocks of symmetric groups of defect at most 5.     

On Abelian Groups Having All Proper Fully Invariant Subgroups Isomorphic

We introduce two classes of abelian groups which have either only trivial fully invariant subgroups or all their nontrivial (respectively nonzero) fully invariant subgroups are isomorphic, called IFI-groups and strongly IFI-groups, such that every strongly IFI-group is an IFI-group, respectively. Moreover, these classes coincide when the groups are torsion-free, but are different when the groups are torsion as well as, surprisingly, mixed groups cannot be IFI-groups. We also study their important properties as our results somewhat contrast with those from [13 Grinshpon , S. Ya. , Nikolskaya (Savinkova) , M. M. ( 2011 ). Fully invariant subgroups of abelian p-groups with finite Ulm-Kaplansky invariants . Commun. Algebra 39 : 4273 – 4282 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] and [14 Grinshpon , S. Ya. , Nikolskaya (Savinkova) , M. M. ( 2011-2012/2014 ). Torsion IF-groups . Fundam. Prikl. Mat. 17 : 47 – 58 ; translated in J. Math. Sci. 197:614–622 . [Google Scholar]].     

Replacement of Factors by Subgroups in the Factorization of Abelian Groups

The above-titled paper of mine appeared in the Bulletin of theLondon Mathematical Society, 32 (2000) 297–304. Regrettably,there is a careless error in the proofs of Theorems 6 and 8.In line 6 of the proof of Theorem 6, it is claimed that a certainsubset must be a subgroup. For this to hold, the subset mustcontain the zero element. This need not be the case; the truededuction is that the subset is a coset, say M + h, of a subgroupM. Now M and M + h contain the same number of elements, andso the deduction that M has p elements is still correct. Similarly, in the proof of Theorem 8, the subgroup M_k must bereplaced by a coset M_k + h_k. This is the only change neededin this proof, since the sum M_k+h_k+(nBH) being direct impliesthat the sum M_k+(nBH) is also direct. Since the zero elementdoes belong to the sets (mAH) and (nBH), the statements aboutthese sets are correct. So the second paragraph of the Proofof Theorem 8 is correct, and is also a proof of Theorem 6. Now we present an example that, we hope, will clarify the situation,as well as showing that certain statements in the original ‘Proof’of Theorem 6 not only could be wrong but actually are wrong.The smallest numerical example occurs with p = 2, m = 3, n =5. Then G is a cyclic group of order 60, and may be representedas the integers modulo 60. Let A = {0, 1, 2, 3, 4} + {0, 15} and B = {0, 5, 10} + {0, 30}.It is easily verified that A + B = {0, 1,..., 59}. In the notationof Theorem 6, we see that H = {0, 15, 30, 45}, K = {0, 12, 24,36, 48}, L = {0, 20, 40}, and M = {0, 30}. Now we see that mA= {0, 3, 6, 9, 12} + {0, 45} M + K, and that nB = {0, 25, 50}+ {0, 30} M + L. We note, however, that A is a complete setof residues modulo 10; that is, that B can be replaced by M+ L.     

Replacement of Factors By Subgroups in the Factorization of Abelian Groups

In his book Abelian groups, L. Fuchs raised the question asto whether, in general, in the factorization of a finite (cyclic)abelian group one factor may always be replaced by some subgroup.The answer turned out to be negative in general, but positivein certain cases. In this paper the complete answer for cyclicgroups is given. In all previously unsolved cases, the answerturns out to be positive. It is shown that a cyclic group hasthe property that in every factorization, one factor may bereplaced by a subgroup if and only if the group has order equalto the product of a prime and a square-free integer. Certainresults are also given in non-cyclic cases. 1991 MathematicsSubject Classification 20K01.     

换位子群是不可分Abel群的有限秩可除幂零群

完整地确定了换位子群是不可分Abel群的有限秩可除幂零群的结构,证明了下面的定理.设G是有限秩的可除幂零群,则G的换位子群是不可分Abel群当且仅当G'=Q或Q_p/Z且G可以分解为G=S×D,其中当G'=Q时,■当G'=Q_p/Z时,S有中心积分解S=S_1*S_2*…*S_r,并且可以将S形式化地写成■其中■,式中s,t都是非负整数,Q是有理数加群,π_κ(k=1,2,…,t)是某些素数的集合,满足π_1■Cπ_2■…■π_t,Q_π_k={m/n|(m,n)=1,m∈Z,n为正的π_k-数}.进一步地,当G'=Q时,(r;s;π_1,π_2,…,π_t)是群G的同构不变量;当G'=Q_p/Z时,(p,r;s;π_1,π_2,…,πt)是群G的同构不变量.即若群H也是有限秩的可除幂零群,它的换位子群是不可分Abel群,那么G同构于H的充分必要条件是它们有相同的不变量.     

IA-Automorphisms of Metabelian Products of Two Abelian Groups

For metabelian products of two Abelian groups with torsion, IA-automorphisms that are not inner and hence are not induced by any automorphisms of the free products of the same Abelian groups are constructed.