Let $G$ be a finite $p$-group of order $p^{n}$ with $|M(G)|=p^{\frac{n(n-1)}{2}-t},$ where $M(G)$ is the Schur multiplier of $G$. Ya.G. Berkovich, X. Zhou, and G. Ellis have determined the structure of $G$ when $t=0,1,2,3$. In this paper, we are going to find some structures for an abelian $p$-group $G$ with conditions on the exponents of $G, M(G),$ and $S_2M(G)$, where $S_2M(G)$ is the metabelian multiplier of $G$.
We give a bound on the order of the Schur multiplier of $p$-groups refining earlier bounds. As an application we complete the classification of groups having Schur multiplier of maximum order. Finally we prove that the order of the Schur multiplier of a finite $p$-group of maximal class and order $p^n$ is at most $p^{\frac{n}{2}}$.
The main aim of the article is to find the Schur multiplier and the Lie exterior square of some finite multiplicative Lie algebras. For a non abelian simple group $K$ with trivial Schur multiplier, we see that the Schur multiplier of multiplicative Lie algebra $K$ is trivial and the Lie exterior square of $K$ is an improper multiplicative Lie algebra $K.$
We introduce the $q$-Bogomolov multiplier as a generalization of the Bogomolov multiplier, and we prove that it is invariant under $q$-isoclinism. We prove that the $q$-Schur Multiplier is invariant under $q$- exterior isoclinism, and as an easy consequence we prove that the Schur multiplier is invariant under exterior isoclinism. We also prove that if $G$, $H$ are $p$-groups and $G/Z^{\wedge}(G)\cong H/Z^{\wedge}(H)$, then the cardinality of the minimal number of generators of $G$ and $H$ are the same. Moreover we prove some structural results about $q$-nonabelian tensor square of groups.
We describe an algorithm for computing successive quotients of the Schur multiplier $M(G)$ for a group $G$ given by an invariant finite $L$-presentation. As application, we investigate the Schur multipliers of various self-similar groups such as the Grigorchuk super-group, the generalized Fabrykowski-Gupta groups, the Basilica group and the Brunner-Sidki-Vieira group.
An explicit formula for the Schur multiplier of the group of unitriangular matrices over products of cyclic rings $\ZZ/m\ZZ$ and $\ZZ$ is derived. We use it to provide presentations of the corresponding covering groups and touch upon the question of finding their nonisomorphic representatives.
In this article, we prove that the Schur Multiplier of a finite $p$-group of maximal class of order $p^n ~(4 \leq n \leq p+1)$ is elementary abelian. The case $n = p+1$ settles a question raised by Primož Moravec in an earlier article.
In this article, we compute the Schur multiplier, non-abelian tensor square and exterior square of non-abelian $p$-groups of order $p^5$. As an application we determine the capability of groups of order $p^5$.
We describe an algorithm to compute the Schur multipliers of all nilpotent Lie $p$-rings in the family defined by a symbolic nilpotent Lie $p$-ring. Symbolic nilpotent Lie $p$-rings can be used to describe the isomorphism types of $p$-groups of order $p^n$ for $n \leq 7$ and all primes $p \geq n$. We apply our algorithm to compute the Schur multipliers of all $p$-groups of order dividing $p^6$.
In this paper, we find an upper bound for the exponent of the Schur multiplier of a pair $(G,N)$ of finite $p$-groups, when $N$ admits a complement in $G$. As a consequence, we show that the exponent of the Schur multiplier of a pair $(G,N)$ divides $\exp(N)$ if $(G,N)$ is a pair of finite $p$-groups of class at most $p-1$. We also prove that if $N$ is powerfully embedded in $G$, then the exponent of the Schur multiplier of a pair $(G,N)$ divides $\exp(N)$.
In 1998, G. Ellis defined the Schur multiplier of a pair $(G,N)$ of groups and mentioned that this notion is a useful tool for studying pairs of groups. In this paper, we characterize the structure of a pair of finite $p$-groups $(G,N)$ in terms of the order of the Schur multiplier of $(G,N)$ under some conditions.
A group G is called special p-group of rank k if the commutator subgroup [G,G] and centre Z(G) are equal, which is elementary abelian p-group of rank k and G/[G,G] is also elementary abelian p-group. In this article we determine the Schur multiplier of special p-groups of rank 2 explicitly.
Let $G$ be a finite $p$-group of order $p^n$ and $M(G)$ be its Schur multiplier. It is well known result by Green that $|M(G)|= p^{\frac{1}{2}n(n-1)-t(G)}$ for some $t(G) \geq 0$. In this article we classify non-abelian $p$-groups $G$ of order $p^n$ for $t(G)=\log_p(|G|)+1$.
The aim of this paper is to determine the non-abelian tensor square and Schur multiplier of groups of square free order and of groups of orders $p^2q$, $pq^2$ and $p^2qr$, where $p$, $q$ and $r$ are primes and $p<q<r$.
Let $G$ be a non-abelian $p$-group of order $p^n$ and $M(G)$ be its Schur multiplier. It is well known result by Green that $|M(G)| \leq p^{\frac{1}{2}n(n-1)}$. So $|M(G)|= p^{\frac{1}{2}n(n-1)-t(G)}$ for some $t(G) \geq 0$. The groups has already been classified for $t(G) \leq 5$ by several authors. For $t(G)=6$ the classification has been done. In this paper we classify $p$-groups $G$ for $t(G) = 6$ in different method.
In 1956, Green provided a bound on the order of the Schur multiplier of $p$-groups. This bound, given as a function of the order of the group, is the best possible. Since then, the bound has been refined numerous times by adding other inputs to the function, such as, the minimal number of generators of the group and the order of the derived subgroup. We strengthen these bounds by adding another input, the group's nilpotency class. The specific cases of nilpotency class 2 and maximal class are discussed in greater detail.
In this note, we give short proofs of the well-known results that the exponent of the Schur multiplier $\M$ divides the exponent of $\G$ for finite $\p$-groups of maximal class and potent $\p$-groups. Moreover, we prove the same for a finite $\p$-group $\G$ satisfying $\G^{\p^2}\subset γ_{\p}(\G)$, and for $3$-groups of class $5$. We do this by proving a general lemma, and show that these three classes of groups satisfy the hypothesis of our lemma.
A longstanding problem attributed to I. Schur says that for a finite group $G$, the exponent of the second homology group $H_2(G, \mathbb{Z})$ divides the exponent of $G$. In this paper, we prove this conjecture for finite nilpotent groups of odd exponent and of nilpotency class 5, $p$-central metabelian $p$ groups, and groups considered by L. E . Wilson in \cite{LEW}. Moreover, we improve several bounds given by various authors.
Let $d(G)$ be the minimum number of elements required to generated a group $G.$ For a group $G $ of order $p^n$ with derived subgroup of order $ p^k $ and $d(G) = d,$ we knew the order of the Schur multiplier of $G$ is bounded by $ p^{\frac{1}{2}(d-1)(n-k+2)+1}. $ In the current paper, we find the structure of all $p$-groups that attains the mentioned bound. Moreover, we show that all of them are capable.
There is a long-standing conjecture attributed to I Schur that if $G$ is a finite group with Schur multiplier $M(G)$ then the exponent of $M(G)$ divides the exponent of $G$. It is easy to see that this conjecture holds for exponent 2 and exponent 3, but it has been known since 1974 that the conjecture fails for exponent 4. In this note I give an example of a group $G$ with exponent 5 with Schur multiplier $M(G)$ of exponent 25, and an example of a group $A$ of exponent 9 with Schur multiplier $M(A)$ of exponent 27.
Let $G$ be a finite $p$-group of order $p^n$. It is known that $|\mathcal{M}(G)|=p^{\f{1}{2}n(n-1)-t(G)}$ and $t(G)\geq 0$. The structure of $G$ characterized when $t(G)\leq 4$ in \cite{be,el,ni,sa,zh}. The structure description of $G$ is determined in this paper for $t(G)=5$.
In this paper, using a relationship between the Schur multiplier of a group $G$, the fundamental group, and the second homology group of the Eilenberg-MacLane space of $G$, we present new proofs for some famous properties of the Schur multiplier and its structure for the free product and the direct product with an algebraic topological approach. Also, we try to find the Schur multiplier of a free amalgamated product in terms of Schur multipliers of its factors, which is a new result.
For a finite group we introduce a particular central extension, the unitary cover, having minimal exponent among those satisfying the projective lifting property. We obtain new bounds for the exponent of the Schur multiplier relating to subnormal series, and we discover new families for which the bound is the exponent of the group. Finally, we show that unitary covers are controlled by the Zel'manov solution of the restricted Burnside problem for 2-generator groups.
Liedtke (2008) has introduced group functors $K$ and $\tilde K$, which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are connected to the theory of central extensions and Schur multipliers. In this work we relate $K$ and $\tilde K$ to a group functor $τ$ arising in the construction of the non-abelian exterior square of a group. In contrast to $\tilde K$, there exist efficient algorithms for constructing $τ$, especially for polycyclic groups. Supported by computations with the computer algebra system GAP, we investigate wh…
It has been proved in \cite{ge} for every $p$-group of order $p^n$, $|\mathcal{M}(G)|=p^{\f{1}{2}n(n-1)-t(G)}$, where $t(G)\geq 0$. In \cite{be, el, zh}, the structure of $G$ has been characterized for $t(G)=0,1,2,3$ by several authors. Also in \cite{sa}, the structure of $G$ characterized when $t(G)=4$ and $Z(G)$ is elementary abelian. This paper is devoted to classify the structure of $G$ when $t(G)=4$ without any condition.
Let $G$ be a central product of two groups $H$ and $K$. We study second cohomology group of $G$, having coefficients in a divisible abelian group $D$ with trivial $G$-action, in terms of the second cohomology groups of certain quotients of $H$ and $K$. In particular, for $D = \mathbb{C}^{*}$, some of our results provide a refinement of results from [Some groups with non-trivial multiplicators, Math. Z. {\bf 120 } (1971), 307-308] and [On the Schur multiplicator of a central quotient of a direct product of groups, J. Pure Appl. Algebra {\bf 3} (1973), 73-82].
Let $G$ be a finite $p$-group and $N$ be a normal subgroup of $G$, with $|N|=p^n$ and $|G/N|=p^m$. A result of Ellis (1998) shows that the order of the Schur multiplier of such a pair $(G,N)$ of finite $p$-groups is bounded by $ p^{\frac{1}{2}n(2m+n-1)}$ and hence it is equal to $ p^{\frac{1}{2}n(2m+n-1)-t}$, for some non-negative integer $t$. Recently the authors characterized the structure of $(G,N)$ when $N$ has a complement in $G$ and $t\leq 3$. This paper is devoted to classify the structure of $(G,N)$ when $N$ has a normal complement in $G$ and $t=4,5$.
Let $G$ be a non-abelian $p$-group of order $p^n$ and $M(G)$ denote the Schur multiplier of $G$. Niroomand proved that $|M(G)| \leq p^{\frac{1}{2}(n+k-2)(n-k-1)+1}$ for non-abelian $p$-groups $G$ of order $p^n$ with derived subgroup of order $p^k$. Recently Rai classified $p$-groups $G$ of nilpotency class $2$ for which $|M(G)|$ attains this bound. In this article we show that there is no finite $p$-group $G$ of nilpotency class $c \geq 3$ for $p\neq3$ such that $|M(G)|$ attains this bound. Hence $|M(G)| \leq p^{\frac{1}{2}(n+k-2)(n-k-1)}$ for $p$-groups $G$ of class $c \geq 3$ where $p \neq 3…
In this paper, using a relation between Schur multipliers of pairs and triples of groups, the fundamental group and homology groups of a homotopy pushout of Eilenberg-MacLane spaces, we present among other things some behaviors of Schur multipliers of pairs and triples with respect to free, amalgamated free, and direct products and also direct limits with topological approach.
In this article, we study the Schur mutiplier of the discrete as well as the finite Heisenberg groups and their t-variants. We describe the representation groups of these Heisenberg groups and through these give a construction of their finite dimensional complex projective irreducible representations.
We fix a finitely presented group $Q$ and consider short exact sequences $1\to N\to G\to Q\to 1$ with $G$ finitely generated. The inclusion $N\to G$ induces a morphism of profinite completions $\hat N\to \hat G$. We prove that this is an isomorphism for all $N$ and $G$ if and only if $Q$ is super-perfect and has no proper subgroups of finite index. We prove that there is no algorithm that, given a finitely presented, residually finite group $G$ and a finitely presentable subgroup $P\subset G$, can determine whether or not $\hat P\to\hat G$ is an isomorphism.
Let $G$ be a $p$-group of nilpotency class $k$ with finite exponent $\exp(G)$ and let $m=\lfloor\log_pk\rfloor$. We show that $\exp(M^{(c)}(G))$ divides $\exp(G)p^{m(k-1)}$, for all $c\geq1$, where $M^{(c)}(G)$ denotes the c-nilpotent multiplier of $G$. This implies that $\exp(M(G))$ divides $\exp(G)$ for all finite $p$-groups of class at most $p-1$. Moreover, we show that our result is an improvement of some previous bounds for the exponent of $M^{(c)}(G)$ given by M. R. Jones, G. Ellis and P. Moravec in some cases.
In this paper we obtain an explicit formula for the higher Schur-multiplicator of an arbitrary finite abelian group with respect to the variety of nilpotent groups of class at most $c\geq 1$ .
We give a contractive Schur multiplier characterization of locally compact groups coarsely embeddable into Hilbert spaces. Consequently, all locally compact groups whose weak Haagerup constant is 1 embed coarsely into Hilbert spaces, and hence the Baum-Connes assembly map with coefficients is split-injective for such groups.
We define a map from second quandle homology to the Schur multiplier and examine its properties. Furthermore, we express the second homology of Alexander quandles in terms of exterior algebras. Additionally, we present a self-contained proof of its structure and provide some computational examples.
The author in $($On the order of Schur multiplier of non-abelian $p$-groups. J. Algebra (2009).322: 4479--4482$)$ showed that for any $p$-group $G$ of order $p^n$ there exists a nonnegative integer $s(G)$ such that the order of Schur multiplier of $G$ is equal to $p^{\f{1}{2}(n-1)(n-2)+1-s(G)}$. Furthermore, he characterized the structure of all non-abelian $p$-groups $G$ when $s(G)=0$. The present paper is devoted to characterization of all $p$-groups when $s(G)=2$.
Let X be a homogeneous tree of degree q+1 (for q between 2 and infinity) and let f be a complex function on X times X for which f(x,y) only depend on the distance between x and y in X. Our main result gives a necessary and sufficient condition for such a function to be a Schur multiplier on X times X. Moreover, we find a closed expression for the Schur norm of f. As applications, we obtain a closed expression for the completely bounded Fourier multiplier norm of the radial functions on the free (non-abelian) group on N generators (for N between 2 and infinity) and of the spherical functions on…
Schur's theorem and its relation to the closure properties of the non-abelian tensor product
Universidade de Brasilia（巴西利亚大学）；Vladimir Chavchanidze Institute of Cybernetics of the Georgian Technical University（格鲁吉亚理工大学弗拉基米尔·查夫奇尼泽网络学研究所）；University of Santiago de Compostela（圣地亚哥德孔波斯特拉大学）
Let $G$ be a finite $p$-group of order $p^n$ with $|G'| = p^k$. Let $M(G)$ denotes the Schur multiplier of $G$. A classical result of Green states that $|M(G)| \leq p^{\frac{1}{2}n(n-1)}$. In 2009, Niroomand, improving Green's and other bounds on $|M(G)|$ for a non-abelain $p$-group $G$, proved that $|M(G)| \leq p^{\frac{1}{2}(n-k-1)(n+k-2)+1}$. In this article we note that a bound, obtained earlier, by Ellis and Weigold is more general than the bound of Niroomand. We derive from the bound of Ellis and Weigold that $|M(G)| \leq p^{\frac{1}{2}(d(G)-1)(n+k-2)+1}$ for a non-abelain $p$-group $G$.…
We exhibit an explicit construction for the second cohomology group $H^2(L, A)$ for a Lie ring $L$ and a trivial $L$-module $A$. We show how the elements of $H^2(L, A)$ correspond one-to-one to the equivalence classes of central extensions of $L$ by $A$, where $A$ now is considered as an abelian Lie ring. For a finite Lie ring $L$ we also show that $H^2(L, \C^*) \cong M(L)$, where $M(L)$ denotes the Schur multiplier of $L$. These results match precisely the analogue situation in group theory.
In order to investigate the relationship between weak amenability and the Haagerup property for groups, we introduce the weak Haagerup property, and we prove that having this approximation property is equivalent to the existence of a semigroup of Herz-Schur multipliers generated by a proper function. It is then shown that a (not necessarily proper) generator of a semigroup of Herz-Schur multipliers splits into a positive definite kernel and a conditionally negative definite kernel. We also show that the generator has a particularly pleasant form if and only if the group is amenable. In the sec…
In this note, starting with any group homomorphism $f\colonΓ\to G$, which is surjective upon abelianization, we construct a universal central extension $u\colon U\twoheadrightarrow G,$ UNDER $Γ$ with the same surjective property, such that for any central extension $m\colon M\twoheadrightarrow G,$ under $f,$ there is a unique homomorphism $U\to M$ with the obvious commutation condition. The kernel of $u$ is the relative Schur multiplier group $H_2(G,Γ;\mathbb{Z})$ as defined in the paper. The case where $G$ is perfect corresponds to $Γ=1$. This yields homological obstructions to lifting soluti…
Primoz Moravec published a very important paper in 2007 where he proved that if $G$ is a finite group of exponent $n$ then the exponent of the Schur multiplier of $G$ can be bounded by a function $f(n)$ depending only on $n$. Moravec does not give a value for $f(n)$, but actually his proof shows that we can take $f(n)=ne$ where $e$ is the order of $b^{-n}a^{-n}(ab)^{n}$ in the Schur multiplier of $R(2,n)$. (Here$R(2,n)$ is the largest finite two generator group of exponent $n$, and we take $a,b$ to be the generators of $R(2,n)$.) It is an easy hand calculation to show that $e=n$ for $n=2,3$, a…
We prove that for a finitely generated group $G$, the second stable homotopy group $π_2^S(K(G,1))$ of the Eilenberg-Maclane space $K(G,1)$ is completely determined by the Schur multiplier $H_2(G)$. We also prove that the second stable homotopy group $π_2^S(K(G,1))$ is equal to the Schur multiplier $H_2(G)$ for a torsion group $G$ with no elements of order $2$ and show that for such groups, $π_2^S(K(G,1))$ is a direct factor of $π_{3}(SK(G,1))$, where $S$ denotes suspension and $π_2^S$ the second stable homotopy group. We compute $π_{3}(SK(G,1))$ and $π_2^S(K(G,1))$ for symmetric, alternating, …
Recently Rai obtained an upper bound for the order of the Schur multiplier of a $d$-generator special $p$-group when its derived subgroup has the maximum value $ p^{\frac{1}{2}d(d-1)}$ for $ d\geq 3 $ and $ p\neq 2. $ Here we try to obtain the Schur multiplier, the exterior square and the tensor square of such $p$-groups. Then we specify which ones are capable. Moreover, we give an upper bound for the order of the Schur multiplier, the exterior product and the tensor square of a $d$-generator special $p$-group $ G $ when $ |G'|=p^{\frac{1}{2}d(d-1)-1}$ for $ d\geq 3 $ and $ p\neq 2. $ Addition…
A Schur ring over a finite group is said to be decomposable if it is the generalized wreath product of Schur rings over smaller groups. In this paper we establish a sufficient condition for a decomposable Schur ring over the direct product of elementary abelian groups to be a $CI$-Schur ring. By using this condition we reprove in a short way known results on the $CI$-property for decomposable Schur rings over an elementary abelian group of rank at most $5$.
We extend the notion of free $p$-central groups for odd primes $p$ to the case $p=2$ by introducing a variant of the lower $p$-central series. This enables us to calculate Schur multipliers of free $p$-central groups. We also prove that for any $p$-central group the exponent of its Schur multiplier divides the exponent of the group, and determine its exponential rank.
A finite group $G$ is called a Schur group if any Schur ring over $G$ is associated in a natural way with a subgroup of $Sym(G)$ that contains all right translations. It is proved that the group $C_3\times C_3\times C_p$ is Schur for any prime $p$. Together with earlier results, this completes a classification of the abelian Schur groups of odd order.
We prove that the class of nilpotent by finite, solvable by finite, polycyclic by finite, nilpotent of nilpotency class $n$ and supersolvable groups are closed under the formation of the non-abelian tensor product. We provide necessary and sufficient conditions for the non-abelian tensor product of finitely generated groups to be finitely generated. We prove that central extensions of most finite simple groups have trivial Bogomolov multiplier.
This article focuses on those aspects about partial actions of groups which are related to Schur's theory on projective representations. It provides an exhaustive description of the partial Schur multiplier, and this result is achieved by introducing the concept of a second partial cohomology group relative to an ideal, together with an appropriate analogue of a central extension. In addition, the new framework is proved to be consistent with the earlier notion of cohomology over partial modules.
