Igor Ya. Subbotin
The article on the upper central series of infinite groups by M. de Falco, F. de Giovanni, C. Musella and Y.P. Sysak, proceedings of the american mathematical society, Volume 139, Number 2, February 2011, 385--389 consists of a quite long proof of a main theorem. We offer a simple and very brief proof of this result.
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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 $G$.…
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…
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.
Notes on $B$-groups
本文通过区分 $B$-群与 $BS$-群，构造了无穷多个 $B$-群但非 $BS$-群的例子，揭示了 Schur 环方法在原始群研究中的局限性。
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.
Let $G$ be an arbitrary group such that $G/\Z(G)$ is finite, where $\Z(G)$ denotes the center of the group $G$. Then $γ_2(G)$, the commutator subgroup of $G$, is finite. This result is known as Shur's theorem (the Schur's theorem). In this short note we provide a quick survey on the converse of Schur's theorem, generalize known results in this direction and prove the following result (which is perhaps the most suitable statement for converse of the Schur's theorem): If $G$ is an arbitrary group with finite $γ_2(G)$, then $G/\Z(G)$ is finite if $\Z_2(G)/\Z(\Z_2(G))$ is finitely generated, where…
These are the notes for some lectures given by this author at Harish-Chandra Research Institute, Allahabad in March 2014 for a workshop on Schur multipliers. The lectures aimed at giving an overview of the subject with emphasis on groups of Lie type over finite, real and $p$-adic fields.
W.Haebich (Bull. Austral. Math. Soc., 7, 1972, 279-296) presented a formula for the Schur multiplier of a regular product of groups. In this paper first, it is shown that the Baer-invariant of a nilpotent product of groups with respect to the variety of nilpotent groups has a homomorphic image and in finite case a subgroup of Haebich's type. Second a formula will be presented for the Baer-invariant of a nilpotent product of cyclic groups with respect to the variety of nilpotent groups.
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…
The Schur--Zassenhaus Theorem and Sylow's Third Theorem for Finite Skew Braces
本文证明了有限斜支架的Schur-Zassenhaus定理和Sylow第三定理，包括Hall理想的子斜支架补的存在性以及Sylow p-子斜支架个数模p同余1的性质。
We present new results regarding automatic continuity, unifying some diagonalization concepts that have been developed over the years. For example, any homomorphism from a completely metrizable topological group to Thompson's group $F$ has open kernel. A similar claim holds when $F$ is replaced with a Baumslag-Solitar group or a torsion-free word hyperbolic group.
In 1904, Issai Schur proved the following result. If $G$ is an arbitrary group such that $G/\Z(G)$ is finite, where $\Z(G)$ denotes the center of the group $G$, then the commutator subgroup of $G$ is finite. A partial converse of this result was proved by B. H. Neumann in 1951. He proved that if $G$ is a finitely generated group with finite commutator subgroup, then $G/\Z(G)$ is finite. In this short note, we exhibit few arguments of Neumann, which provide further generalizations of converse of the above mentioned result of Schur. We classify all finite groups $G$ such that $|G/\Z(G)| = |γ_2(G…
A conjecture regarding the structure of expander graphs is discussed.
Some answers regarding factorizations of finite groups
针对《库罗夫卡笔记本》中关于有限群分解的问题，G. M. 伯格曼给出\(k = 3\)反例并提问，本文给出\(k \geq 3\)的反例，解决了部分问题，\(k = 2\)时问题仍待解决。
We shortly present two small results regarding the study formal duality in finite abelian groups as introduced by Cohn, Kumar, Reiher and Schürmann. In particular, we give a new example of a formally self dual set in $\mathbb{Z}_2^2\times\mathbb{Z}_4^2$ and computed nonexistence of primitive formally dual sets of size $8$ in $\mathbb{Z}_8^2$.
An answer regarding automorphisms of finite abelian groups
本文否定了一个关于有限阿贝尔群自同构群大小与群大小比值的猜想，证明该比值为有理数时分母必须平方自由，并且奇素数不能作为此类比值。
In this note we describe a seemingly new approach to the complex representation theory of the wreath product $G\wr S_d$ where $G$ is a finite abelian group. The approach is motivated by an appropriate version of Schur-Weyl duality. We construct a combinatorially defined groupoid in which all endomorphism algebras are direct products of symmetric groups and prove that the groupoid algebra is isomorphic to the group algebra of $G\wr S_d$. This directly implies a classification of simple modules. As an application, we get a Gelfand model for $G\wr S_d$ from the classical involutive Gelfand model …
In this note we describe the role of the Schur multiplier in the structure of the $p$-torsion of discrete groups. More concretely, we show how the knowledge of $H_2G$ allows to approximate many groups by colimits of copies of finite $p$-groups. Our examples include interesting families of non-commutative infinite groups, including Burnside groups, certain solvable examples and the first Grigorchuk group. We also provide a counterexample for a conjecture of E. Farjoun.
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