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For a nilpotent group G without π -torsion, and x , y ∈ G , if x n = y n for a π -number n , then x = y ; if x m y n = y n x m for π -numbers m , n , then x y = y x. This is a well-known result in group theory. In this paper, we prove two analogous theorems on matrices, which have independence significance. Specifically, let m be a given positive integer and A a complex square matrix satisfying that (i) all eigenvalues of A are nonnegative, and (ii) rank ⁡ A 2 = rank ⁡ A ; then A has a unique m -th root X with rank ⁡ X 2 = rank ⁡ X , all eigenvalues of X are nonnegative, and moreover there is a polynomial f (λ) with X = f (A). In addition, let A and B be complex n × n matrices with all eigenvalues nonnegative, and rank ⁡ A 2 = rank ⁡ A , rank ⁡ B 2 = rank ⁡ B ; then (i) A = B when A r = B r for some positive integer r , and (ii) A B = B A when A s B t = B t A s for two positive integers s and t. [ABSTRACT FROM AUTHOR]

A subset D of a group G is a determining set of G if every automorphism of G is uniquely determined by its action on D , and the determining number of G , α (G) , is the cardinality of a smallest determining set. A group G is called a DEG-group if α (G) equals γ (G) , the generating number of G. Our main results are as follows. Finite groups with determining number 0 or 1 are classified; finite simple groups and finite nilpotent groups are proved to be DEG-groups; for a given finite group H , there is a DEG-group G such that H is isomorphic to a normal subgroup of G and there is an injective mapping from the set of all finite groups to the set of finite DEG-groups; for any integer k ≥ 2 , there exists a group G such that α (G) = 2 and γ (G) ≥ k. [ABSTRACT FROM AUTHOR]

Suppose that L is a Lie superalgebra over a field F of characteristic different from 2 and 3. In this paper, the so-called 5-sequence of cohomology for a central extension of L is constructed and is proved to be exact. Moreover, the multiplier of L is proved to be isomorphic to the second cohomology group with coefficients in the trivial module of L. Finally, an upper bound of the superdimension of the second cohomology group is given in the situation when L is nilpotent and finite-dimensional. [ABSTRACT FROM AUTHOR]

We obtain a complete characterization of the structure of a finite group G in which every maximal subgroup is nilpotent or a TI-subgroup or has order p ′ for any fixed prime divisor p of | G |. Moreover, we show that there exists at most one prime divisor q of | G | such that G is neither q -nilpotent nor q -closed. [ABSTRACT FROM AUTHOR]

Let A be a completely decomposable homogeneous torsion-free abelian group of rank n (n ≥ 2). Let Θ m (n) = A ⋊ ⟨ α ⟩ be the split extension of A by an automorphism α which is a cyclic permutation of the direct components twisted by a rational integer m. Then Θ m (n) is an infinite soluble group. In this paper, the residual finiteness of Θ m (n) is investigated. [ABSTRACT FROM AUTHOR]

Let A be a completely decomposable homogeneous torsion-free abelian group of rank n (n ≥ 2). Let Θ m (n) = A ⋊ ⟨ α ⟩ be the split extension of A by an automorphism α which is a cyclic permutation of the direct components twisted by a rational integer m. Then Θ m (n) is an infinite soluble group. In this paper, the residual finiteness of Θ m (n) is investigated. [ABSTRACT FROM AUTHOR]

We obtain a complete characterization of the structure of a finite group G in which every maximal subgroup is nilpotent or a TI-subgroup or has order p ′ for any fixed prime divisor p of | G |. Moreover, we show that there exists at most one prime divisor q of | G | such that G is neither q -nilpotent nor q -closed. [ABSTRACT FROM AUTHOR]

Let F, N, A and N2 denote the properties of being finite, nilpotent, abelian and nilpotent of classes at most 2, respectively. Firstly we consider the class of finitely generated FN-groups. We show that the property FC is closed under finite extensions, and extend this result to finitely generated NF-groups. Secondly we prove that a finitely generated NF-group G is in the class ((FC)F, ∞) if and only if G is an FA-group. Finally we prove that a finitely generated NF-group in the class ((FC)F, ∞)* is an FN2-group. Moreover, G/Z2(G) is finite. [ABSTRACT FROM AUTHOR]

Let K be a 2-torsion free ring with identity and Rn(K, J) be the ring of all n × n matrices over K such that the entries on and above the main diagonal are elements of an ideal J of K. We describe all Jordan derivations of the matrix ring Rn(K, J) in this paper. The main result states that every Jordan derivation Δ of Rn(K, J) is of the form Δ = D + Ω, where D is a derivation of Rn(K, J) and Ω is an extremal Jordan derivation of Rn(K, J). [ABSTRACT FROM AUTHOR]

A finite group G is called a special local 2-nilpotent group if G is not 2-nilpotent, the Sylow 2-subgroup P of G has a section isomorphic to the quaternion group of order 8, Ω 1   (P   ∩   G ′)   ≤   Z (P) and NG(P) is 2-nilpotent. In this paper, it is shown that SL2(q), q   >   3 , is a special local 2-nilpotent group if and only if q 2 ≡ 1   (mod 16) , and that GL2(q), q > 3 , is a special local 2-nilpotent group if and only if q is odd. Moreover, the solvability of finite groups is also investigated by giving two generalizations of a result from [A note on p-nilpotence and solvability of finite groups, J. Algebra321 (2009) 1555–1560]. [ABSTRACT FROM AUTHOR]

Let G be an extension of a finite quasinilpotent group by a finite group. It is shown that under some conditions every Coleman automorphism of G is an inner automorphism. The interest in such automorphisms arose from the study of the normalizer problem for integral group rings. Our theorems generalize some well-known results. [ABSTRACT FROM AUTHOR]

We solve the quadratic matrix equation AXA = XAX with a given nilpotent matrix A, to find all commuting solutions. We first provide a key lemma, and consider the special case that A has only one Jordan block to motivate the idea for the general case. Our main result gives the structure of all the commuting solutions of the equation with an arbitrary nilpotent matrix. [ABSTRACT FROM AUTHOR]

In this paper, we prove that every local superderivation on basic classical Lie superalgebras except for A(1, 1) over the complex number field ℂ is a superderivation. Furthermore, we give an example of a class of nilpotent Lie superalgebras with local superderivations which are not superderivations. [ABSTRACT FROM AUTHOR]

Let G be a finite group. It is proved that any class-preserving Coleman automorphism of G is an inner automorphism whenever G belongs to one of the following two classes of groups: (1) CN-groups, i.e., groups in which the centralizer of any element is nilpotent; (2) CIT-groups, i.e., groups of even order in which the centralizer of any involution is a 2-group. In particular, the normalizer conjecture holds for both CN-groups and CIT-groups. Additionally, some other results are also obtained. [ABSTRACT FROM AUTHOR]

The authors study and classify finite groups with their automorphism groups having orders 4 pq2, where p and q are primes such that 2 < p < q. In 2013 the first and the third authors classified the nilpotent groups of this kind; here the authors give the classification of those finite non-nilpotent groups with their automorphism groups having orders 4 pq2. [ABSTRACT FROM AUTHOR]

In this paper we prove that any finite nilpotent alternative ring with planar zero-divisor graph is associative. [ABSTRACT FROM AUTHOR]

A group G is said to have property μ whenever N is a non-locally nilpotent normal subgroup of G, there is a finite non-nilpotent G-quotient of N. FC-groups and groups with property ν satisfy property μ, where a group G is said to have property ν if every non-nilpotent normal subgroup of G has a finite non-nilpotent G-quotient. HP(G) is the Hirsch-Plotkin radical of G, and Φf(G) is the intersection of all the maximal subgroups of finite index in G (here Φf(G)=G if no such maximal subgroups exist). It is shown that a group G has property μ if and only if HP(G/Φf(G))=HP(G)/Φf(G) and that the class of groups with property ν is a proper subclass of that of groups with property μ. Also, the structure of the normal subgroups of a group: nilpotency with the minimal condition, is investigated. [ABSTRACT FROM AUTHOR]

The main purpose of this note is to show that there is a one-to-one correspondence between minimal non-nilpotent (resp., locally nilpotent) saturated fusion systems and finite p′-core-free p-constrained minimal non-nilpotent (resp., locally p-nilpotent) groups. [ABSTRACT FROM AUTHOR]

This paper interprets mixed multiplicities of good filtrations as Hilbert-Samuel multiplicities, and shows that (휀1,...,휀m)-superficial sequences in [16] (2007) and superficial sequences in [15] (1973) are weak-(FC)-sequences in [19] (2000). As consequences, we not only obtain generalized results for mixed multiplicities of ideals in [15, 16, 19] but also get an improvement of [19, Theorem 3.4] that seems to account well for the essence of [16, Theorem 1.4]. [ABSTRACT FROM AUTHOR]

In this paper, we describe the finite groups G in which every maximal chain M3 < M2 < M1 < M0 =G contains a proper subnormal subgroup of G. [ABSTRACT FROM AUTHOR]

Let R be a ring with an endomorphism σ. We show that the clean property and various Armendariz-type properties of R are inherited by the skew matrix rings S(R,n,σ) and T(R,n,σ). They allow the construction of rings with a non-zero nilpotent ideal of arbitrary index of nilpotency which inherit various interesting properties of rings. [ABSTRACT FROM AUTHOR]

We prove that a finite group G is p-supersolvable or p-nilpotent if some subgroups of G are weakly s-semipermutable in G. Several earlier results are generalized. [ABSTRACT FROM AUTHOR]

In this paper we give a characterization for certain HNN extensions of subgroup separable groups with normal associated subgroups to be weakly potent. We then apply our result to show that certain HNN extensions of finitely generated nilpotent groups with central associated subgroups are weakly potent. [ABSTRACT FROM AUTHOR]

This paper is devoted to present some properties of the c-nilpotent multiplier Mc(L) and some features of c-central extensions of a (finite dimensional) Lie algebra L. Moreover, we give the structure of all c-covers of Lie algebras whose c-nilpotent multipliers have the Hopfian property. [ABSTRACT FROM AUTHOR]

We consider groups C such that NG(H)/HCG(H) is cyclic for all H < G. More specifically, we characterise locally nilpotent and supersoluble groups with this property. [ABSTRACT FROM AUTHOR]

Let G be a finite group and H a subgroup of G. Recall that H is said to be a TI-subgroup of G if H9∩H = 1 or H for each g ∈G. In this note, we prove that if all non-nilpotent subgroups of a finite non-nilpotent group C are `fl-subgroups, then C is soluble, and all non-nilpotent subgroups of G are normal. [ABSTRACT FROM AUTHOR]

In this paper, we describe the structure of quadratic homogeneous polynomial maps F=X+H with JH3=0. As a consequence we show that in dimension n ≤ 6, JH is strongly nilpotent, or equivalently F=X+H is linearly triangularizable. [ABSTRACT FROM AUTHOR]

Let k be a positive integer. Locally graded groups G for which one of the sets {γk(H)| H ≤ G} and {γk(H)| H ≤G, H} is finite are classified. [ABSTRACT FROM AUTHOR]

Finite nilpotent groups having exactly four conjugacy classes of non-normal subgroups are classified. [ABSTRACT FROM AUTHOR]

Let be the class of groups of non-prime-power order or the class of groups of prime-power order. In this paper we give a complete classification of finite non-solvable groups with a quite small number of conjugacy classes of -subgroups or classes of -subgroups of the same order. [ABSTRACT FROM AUTHOR]

Let be a saturated formation containing , and G be a finite group. Li etc. proposed a problem: whether there is a normality such that the following two statements are equivalent: (i) . (ii) There exists a normal subgroup H of G such that and for each Sylow subgroup P of F*(H), every member in some satisfies the above normality in G. In this paper, we find a normality satisfying the above problem. Moreover, by using the concept of ℋ-subgroups, we obtain other results about the influence of the members of some fixed on the structure of G. [ABSTRACT FROM AUTHOR]

In this paper, a characteristic subgroup of a finite group G is defined, which is the intersection of the normalizers of the nilpotent residuals of all subgroups of G, and the properties of and the relationship between and the group G are investigated. [ABSTRACT FROM AUTHOR]

Suppose that G is a finite group and H is a subgroup of G. We say that H is s-semipermutable in G if HGP = GPH for any Sylow p-subgroup GP of G with (p, |H|) = 1; H is s-semiembedded in G if there is a normal subgroup T of G such that HT is s-semipermutable in G and H ... T < HSSG, where HSAAG is the subgroup of H generated by all those subgroups of H which are s-semipermutable in G. We investigate the influence of s-semiembedded subgroups on the structure of finite groups. Some recent results are generalized and unified [ABSTRACT FROM AUTHOR]

We study the structure of the set of nilpotent elements in extended semicommutative rings and introduce nil α-semicommutative rings as a generalization. We resolve the structure of nil α-semicommutative rings and obtain various necessary or sufficient conditions for a ring to be nil α-semicommutative, unifying and generalizing a number of known commutative-like conditions in special cases. We also classify which of the standard nilpotence properties on polynomial rings pass to skew polynomial ring. Constructing various examples, we classify how the nil α-semicommutative rings behaves under various ring extensions. Also, we consider the nil-Armendariz condition on a skew polynomial ring. [ABSTRACT FROM AUTHOR]

Let G be a finite group. A subgroup H of G is called a 2-maximal subgroup of G if there exists a maximal subgroup M of G such that H is a maximal subgroup of M. In this paper, we discuss the influence of π-quasinormality of 2-maximal subgroups of Sylow subgroups on the structure of a finite group, and obtain some sufficient conditions under which the finite group is p-nilpotent, supersolvable, or possesses an ordered Sylow tower. [ABSTRACT FROM AUTHOR]

The norm N(G) of a group G is the intersection of the normalizes of all the subgroups of G. A group is called capable if it is a central factor group. In this paper, we give a necessary and sufficient condition for a capable group to satisfy N(G)=ζ(G), and then some sufficient conditions for a capable group with N(G)=ζ(G) are obtained. Furthermore, we discuss the norm of a nilpotent group with cyclic derived subgroup. [ABSTRACT FROM AUTHOR]

A subgroup H of a finite group G is called an ℋ-subgroup of G if NG(H) ∩ Hg ≤ H for all g ∈ G. The set of all ℋ-subgroups of a finite group G is denoted by ℋ(G). In this paper, a sufficient condition about p-nilpotency is given and some new results for a finite group G to be p-nilpotent or supersolvable are obtained based on the assumption that some subgroups belong to ℋ(G). [ABSTRACT FROM AUTHOR]

In this paper, we study some properties of the non-abelian tensor product of two groups G and H. More precisely, if G is abelian and H is a nilpotent group, then an upper bound for the exponent of G ⊗ H is obtained. Using our results, we obtain some upper bounds for the exponent of the Schur multiplier of the non-abelian tensor product of groups. Finally, an abelian group is constructed by taking non-abelian tensor product of groups. [ABSTRACT FROM AUTHOR]

In this paper we study flexible algebras (possibly infinite-dimensional) satisfying the polynomial identity x(yz) = y(zx). We prove that in these algebras, products of five elements are associative and commutative. As a consequence of this, we get that when such an algebra is a nil-algebra of bounded nil-index, it is nilpotent. Furthermore, we obtain optimal bounds for the index of nilpotency. Another consequence that we get is that these algebras are associative when they are semiprime. [ABSTRACT FROM AUTHOR]

In this paper we show that some finiteness properties on a centralizer of a particular subgroup can be inherited by the whole group. Among other things, we prove the following characterization of polycyclic groups: a soluble group G is polycyclic if and only if it contains a finitely generated subgroup H, formed by bounded left Engel elements, whose centralizer CG(H) is polycyclic. In the context of Černikov groups we obtain a more general result: a radical group is a Černikov group if and only if it contains a finitely generated subgroup, formed by left Engel elements, whose centralizer is a Černikov group. The aforementioned results generalize a theorem by Onishchuk and Zaĭtsev about the centralizer of a finitely generated subgroup in a nilpotent group. [ABSTRACT FROM AUTHOR]

We introduce a new subgroup embedding property of finite groups called c-normal embedding. By using this embedding property, formation theory and some special techniques, we obtain `iff' versions of theorems of Itô, Buckley, Srinivasan and other related results on p-nilpotence, nilpotence and supersolvability. [ABSTRACT FROM AUTHOR]

Let H, C and D be subgroups of a finite group G. The pair (C, D) is called a θ*-pair for H if (1) D = (C ∩ H)G, 〈Cg, H〉 = G for every g ∈ G, and (2) KH < G for every subgroup K of G with K/D ᐊ G/D and K/D < C/D. In this paper, we obtain some results on θ*-pairs which imply G to be solvable, supersolvable or nilpotent. [ABSTRACT FROM AUTHOR]

Let KG be the group algebra of a group G over a field K of positive characteristic p, and let 픇(n)(G) and 픇[n](G) denote the n-th upper Lie dimension subgroup and the n-th lower one, respectively. In [1] and [12], the equality 픇(n)(G) =픇[n](G) is verified when p ≥ 5. Motivated by [16, Problem 55], in the present paper we establish it for particular classes of groups when p ≤ 3. Finally, we introduce and study a new central series of G linked with the Lie nilpotency class of KG. [ABSTRACT FROM AUTHOR]

A new construction of algebras called a mapping extension of an algebra is here introduced. The construction yields a generalization of some classical constructions such as the nilpotent extension of an algebra, inflation of a semigroup but also the square extension construction introduced recently for idempotent groupoids. The mapping extension construction is defined for algebras of any fixed type, however nullary operation symbols are here not admitted. It is based on the notion of a retraction and some system of mappings. A mapping extension of a given algebra is constructed as a counterimage algebra by a specially defined retraction. Varieties of algebras satisfying an identity φ(x) ≈ x for a term φ not being a variable (such as varieties of lattices, Boolean algebras, groups and rings) are especially interesting because for such a variety $\cal V$, all mapping extensions by φ of algebras from $\cal V$ form an equational class. In the last section, combinatorial properties of the mapping extension construction are considered. [ABSTRACT FROM AUTHOR]

A group G=AB is said to be totally factorized by its subgroups A and B if XY=YX for all subgroups X of A and Y of B. It is known that any finite group totally factorized by supersoluble subgroups is supersoluble, and that a finite group totally factorized by nilpotent subgroups is abelian-by-nilpotent. This latter result is extended here to certain classes of infinite groups. [ABSTRACT FROM AUTHOR]

In 1987, Sullivan characterised the elements of the semigroup NP(X) generated by the nilpotents in P(X), the semigroup (under composition) consisting of all partial transformations of a set X; and in 1999, Marques-Smith and Sullivan determined all the ideals of NP(X) for arbitrary X. In this paper, we use that work to describe all the congruences on NP(X). [ABSTRACT FROM AUTHOR]

Let G be a finite group. A subgroup K of a group G is called an ${\cal H}$-subgroup of G if NG(K) ∩ Kx ≤ K for all x ∈ G. The set of all ${\cal H}$-subgroups of G is denoted by ${\cal H}(G)$. In this paper, we investigate the structure of a group G under the assumption that certain abelian subgroups of prime power order belong to ${\cal H}(G)$. [ABSTRACT FROM AUTHOR]

In this paper we completely study the nilpotency and solubility of the group Autc(G) of central automorphisms of a finite group G. [ABSTRACT FROM AUTHOR]

We study the structure of nilpotent subsemigroups in the semigroup M(n,픽) of all n×n matrices over a field 픽 with respect to the operation of the usual matrix multiplication. We describe the maximal subsemigroups among the nilpotent subsemigroups of a fixed nilpotency degree and classify them up to isomorphism. We also describe isolated and completely isolated subsemigroups and conjugated elements in M(n,픽). [ABSTRACT FROM AUTHOR]

We study the variety of algebras A over a field of characteristic ≠ 2, 3, 5 satisfying the identities xy=yx and β {((xx)y)x-((yx)x)x} + γ {((xx)x)y-((yx)x)x}=0, where β, γ are scalars. We do not assume power-associativity. We prove that if A admits a non-degenerate trace form, then A is a Jordan algebra. We also prove that if A is finite-dimensional and solvable, then it is nilpotent. We find three conditions, any of which implies that a finite-dimensional right-nilalgebra A is nilpotent. [ABSTRACT FROM AUTHOR]