arXiv subjects
Publications and source records attributed to George M. Bergman.
Given a finite group $G$ and a factorization of its order, $|G|= a_1 a_2,$ we show that a sufficient condition for there to exist subsets $A_1,\,A_2\subseteq G$ such that $|A_1| = a_1,$ $|A_2| = a_2,$ and $G = A_1\,A_2,$ is that there exist a chain of subgroups $\{e\} = G_0 < \dots < G_n = G$ such that $a_1$ is the product of some subfamily of the indices $|G_i:G_{i-1}|$ (and hence $a_2$ is the product of the complementary subfamily).
This is a collection of questions for inclusion in future editions of the Kourovka Notebook of open questions in group theory. Most are questions I raised in papers between 1981 and 2021. I welcome feedback.
We recall the concept of the dimension of a finite poset $P$, and the longstanding conjecture that for all finite nonempty posets $P$ and $Q$, $\dim(P\times Q)\geq\dim(P)+\dim(Q)-2.$ We then note two other plausible inequalities, either of which would imply that one. In the final section, writing $P\preccurlyeq P'$ if, for all $Q,$ $\dim(P\times Q)\leq\dim(P'\times Q),$ and writing $P\approx P'$ if $P\preccurlyeq P'$ and $P'\preccurlyeq P,$ we note some results and questions concerning these relations.
Let $P,$ $S,$ and $T$ be semigroups, $f:P\to S$ and $g:P\to T$ semigroup homomorphisms, and $X$ a generating set for $S$ (possibly infinite). Clearly, a necessary condition for there to exist a homomorphism $S\to T$ making a commuting triangle with $f$ and $g$ is that for every relation $f(p) = w(x_1,\,\dots\,,\,x_n)$ holding in $S$, with $p\in P,$ $w$ a semigroup word, and $x_1,\,\dots\,,\,x_n \in X,$ there exist $t_1,\,\dots,\,t_n\in T$ satisfying $g(p) = w(t_1,\,\dots\,,\,t_n).$ Under what assumptions will that also be sufficient? We show that one such family of assumptions is that (i) every element of $S$ is a divisor some element of $f(P),$ (ii) $T$ is right and left cancellative, (iii) $T$ is power-cancellative, i.e, $x^d = y^d \implies x = y$ for $d > 0,$ and (iv) a certain technical condition which, in particular, holds if $T$ admits a semigroup ordering with the order-type of the natural numbers. As an application, we obtain an elementary criterion for the existence of an integer-valued rank function on finitely generated projective modules over a ring.
For $P$ a poset, the dimension of $P$ is defined to be the least cardinal $κ$ such that $P$ is embeddable in a direct product of $κ$ totally ordered sets. We study the behavior of this function on finite-dimensional (not necessarily finite) posets. In general, the dimension dim($P$ x $Q$) of a product of two posets can be smaller than dim($P$) + dim($Q$), though no cases are known where the discrepancy is greater than 2. We obtain a result that gives upper bounds on the dimensions of certain products of posets, including cases where the discrepancy 2 is achieved. But the paper is mainly devoted to stating questions, old and new, about dimensions of product posets, noting implications among their possible answers, and introducing some related concepts that might be helpful in tackling these questions.
If $\mathcal{C}$ is a category of algebras closed under finite direct products, and $M_\mathcal{C}$ the commutative monoid of isomorphism classes of members of $\mathcal{C},$ with operation induced by direct product, A.Tarski defined a nonidentity element $p$ of $M_\mathcal{C}$ to be prime if, whenever it divides a product of two elements in that monoid, it divides one of them, and called an object of $\mathcal{C}$ prime if its isomorphism class has this property. McKenzie, McNulty and Taylor ask whether the category of nonempty semigroups has any prime objects. We show in section 2 that it does not. However, for the category of monoids, and some other subcategories of semigroups, we obtain examples of prime objects in sections 3-4. In section 5, two related questions open so far as I know, are recalled. In section 6, which can be read independently of the rest of this note, we recall two related conditions that are called primeness by semigroup theorists, and obtain results and examples on the relationships among those two conditions and Tarski's, in categories of groups. Section 7 notes an interesting characterization of one of those conditions when applied to finite algebras in an arbitrary variety. Various questions are raised.
Let $ $ be the free monoid on a generating set $X$, and suppose one adjoins to $ $ universal 2-sided inverses to a finite set $S$ of its elements. We note an elementary algorithm which yields a normal form for elements of the resulting monoid $M$. We then show that if $S$ is allowed to be infinite, a similar normal form exists, though it cannot necessarily be computed algorithmically. We raise a couple of questions. We note work by others on the related topic of monoids presented by finite families of relations of the form $w = 1$.
A longstanding open question is whether every strongly clean ring (ring in which every element is strongly clean, i.e., is the sum of an idempotent and a unit which commute with each other) is Dedekind-finite (has the property that every element with a one-sided inverse is invertible). We give an example of a ring with two strongly clean elements that are one-sided, but not two-sided, inverses of one another, suggesting that the answer to that question may be negative. We then discuss possible ways of strengthening this result to give a full negative answer. We end with some brief observations on related topics, in particular, uniquely strongly clean rings.
Let $R$ be a ring, and consider a left $R$-module given with two (generally infinite) direct sum decompositions, $A\oplus(\bigoplus_{i\in I} C_i)=M=B\oplus(\bigoplus_{j\in J} D_j),$ such that the submodules $A$ and $B$ and the $D_j$ are each finitely generated. We show that there then exist finite subsets $I_0\subseteq I,$ $J_0\subseteq J,$ and a direct summand $Y\subseteq \bigoplus_{i\in I_0} C_i,$ such that $A \oplus Y \ =\ B \oplus(\bigoplus_{j\in J_0} D_j).$ We then note some ways that this result can and cannot be generalized, and some related questions.
For which groups $G$ is it true that for all fields $k$, every non-monomial element of the group algebra $k\,G$ generates a proper $2$-sided ideal? The only groups for which we know this are the torsion-free abelian groups. We would like to know whether it also holds for all free groups. It is shown that the above property fails for wide classes of groups: for every group $G$ that contains an element $g\neq 1$ whose image in $G/[g,G]$ has finite order (in particular, every group containing a $g\neq 1$ that itself has finite order, or that satisfies $g\in [g,G])$; and for every group containing an element $g$ which commutes with a conjugate $hgh^{-1}\neq g$ (in particular, for every nonabelian solvable group). Results are obtained on closure properties of the class of groups satisfying the stated condition. Many further questions are raised; in particular, a plausible Freiheitssatz for group algebras of free groups is noted.
We review the definition of a quandle, and in particular of the core quandle $\mathrm{Core}(G)$ of a group $G$, which consists of the underlying set of $G$, with the binary operation $x\lhd y = x y^{-1} x$. This is an involutory quandle, i.e., satisfies the identity $x\lhd (x\lhd y) = y$ in addition to the other identities defining a quandle. Trajectories $(x_i)_{i\in\mathbb{Z}}$ in groups and in involutory quandles (in the former context, sequences of the form $x_i = x z^i$ where $x,z\in G,$ among other characterizations; in the latter, sequences satisfying $x_{i+1}= x_i\lhd\,x_{i-1})$ are examined. A family of necessary conditions for an involutory quandle to be embeddable in the core quandle of a group is noted. Some implications are established between identities holding in groups and in their core quandles. Upper and lower bounds are obtained on the number of elements needed to generate the quandle $\mathrm{Core}(G)$ for $G$ a finitely generated group. Several questions are posed.
F. Wehrung has asked: Given a family $\mathcal{C}$ of subsets of a set $Ω$, under what conditions will there exist a total ordering on $Ω$ under which every member of $\mathcal{C}$ is convex? Note that if $A$ and $B$ are nondisjoint convex subsets of a totally ordered set, neither of which contains the other, then $A\cup B$, $A\cap B$, and $A\setminus B$ are also convex. So let $\mathcal{C}$ be an arbitrary set of subsets of a set $Ω$, and form its closure $\mathcal{P}$ under forming, whenever $A$ and $B$ are nondisjoint and neither contains the other, the sets $A\cup B$, $A\cap B$, and $A\setminus B$. We determine the form $\mathcal{P}$ can take when $\mathcal{C}$, and hence $\mathcal{P}$, is finite, and for this case get necessary and sufficient conditions for there to exist an ordering of $Ω$ of the desired sort. From this we obtain a condition which works without the finiteness hypothesis. We establish bounds on the cardinality of the subset $\mathcal{P}$ generated as above by an $n$-element set $\mathcal{C}$. We note connections with the theory of interval graphs and hypergraphs , which lead to other ways of answering Wehrung's question.
Suppose we wish to embed an (associative) $k$-algebra $A$ in a $k$-algebra $R$ generated in some specified way; e.g., by two elements, or by copies of given $k$-algebras $A_1,$ $A_2,$ $A_3.$ Several authors have obtained sufficient conditions for such embeddings to exist. We prove here some further results on this theme. In particular, we merge the ideas of existing constructions based on two generating elements , and on three given subalgebra , to get a construction using two given subalgebras. We pose some questions on how these results can be further strengthened.
In Question 19.35 of the Kourovka Notebook, M. H. Hooshmand asks whether, given a finite group $G$ and a factorization $\mathrm{card}(G)= n_1\ldots n_k$, one can always find subsets $A_1,\ldots,A_k$ of $G$ with $\mathrm{card}(A_i)=n_i$ such that $G=A_1\ldots A_k;$ equivalently, such that the group multiplication map $A_1\times\ldots\times A_k\to G$ is a bijection. We show that for $G$ the alternating group on 4 elements, $k=3$, and $(n_1,n_2,n_3) = (2,3,2)$, the answer is negative. We then generalize some of the tools used in our proof, and note an open question.
P. M. Cohn showed in 1971 that given a ring $R$, to describe, up to isomorphism, a division ring $D$ generated by a homomorphic image of $R$ is equivalent to specifying the set of square matrices over $R$ which map to singular matrices over $D,$ and he determined precisely the conditions that such a set of matrices must satisfy. The present author later developed another version of this data, in terms of closure operators on free $R$-modules. In this note, we examine the latter concept further, and show how an $R$-module $M$ satisfying certain conditions can be made to induce such data. In an appendix we make some observations on Cohn's original construction, and note how the data it uses can similarly be induced by appropriate sorts of $R$-modules. Our motivation is the longstanding question of whether, for $G$ a right-orderable group and $k$ a field, the group algebra $kG$ must be embeddable in a division ring. Our hope is that the right $kG$-module $M=k((G))$ might induce a closure operator of the required sort. We re-prove a partial result in this direction due to N. I. Dubrovin, note a plausible generalization thereof which would give the desired embedding, and briefly sketch some thoughts on other ways of approaching the problem.
The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper radical ideals (e.g., $\{0\})$ that are closed under the natural metric, but has no prime ideals closed under that metric; hence closed radical ideals are not, in general, intersections of closed prime ideals. Moreover, $B$ is known to be complete in its metric. Together, these facts answer a question posed by J.Gleason. From this example, rings of arbitrary characteristic with the corresponding properties are obtained. The result that $B$ is complete in its metric is generalized to show that if $L$ is a lattice given with a metric satisfying identically either the inequality $d(x\vee y,\,x\vee z)\leq d(y,z)$ or the inequality $d(x\wedge y,\,x\wedge z)\leq d(y,z),$ and if in $L$ every increasing Cauchy sequence converges and every decreasing Cauchy sequence converges, then every Cauchy sequence in $L$ converges; i.e., $L$ is complete as a metric space. We show by example that if the above inequalities are replaced by the weaker conditions $d(x,\,x\vee y)\leq d(x,y),$ respectively $d(x,\,x\wedge y)\leq d(x,y),$ the completeness conclusion can fail. We end with two open questions.
In section 1 we consider a 3-tuple $S=(|S|,\preccurlyeq,E)$ where $|S|$ is a finite set, $\preccurlyeq$ a partial ordering on $|S|,$ and $E$ a set of unordered pairs of distinct members of $|S|,$ and study, as a function of $n\geq 0,$ the number of maps $φ:|S|\to\{1,\dots,n\}$ which are both isotone with respect to the ordering $\preccurlyeq,$ and have the property that $φ(x)\neq φ(y)$ whenever $\{x,y\}\in E.$ We prove a number-theoretic result about this function, and use it in section 7 to recover a ring-theoretic identity of G. P. Hochschild. In section 2 we generalize a result of R. Stanley on the sign-imbalance of posets in which the lengths of all maximal chains have the same parity. In sections 3-6 we study the linearization-count and sign-imbalance of a lexicographic sum of $n$ finite posets $P_i$ $(1\leq i\leq n)$ over an $n$-element poset $P_0.$ We note how to compute these values from the corresponding counts for the given posets $P_i,$ and for a lexicographic sum over $P_0$ of chains of lengths $\mathrm{card}(P_i).$ This makes the behavior of lexicographic sums of chains over a finite poset $P_0$ of interest, and we obtain some general results on the linearization-count and sign-imbalance of these objects.
Marek Kuczma asked in 1980 whether for every positive integer $n,$ there exists a subsemigroup $M$ of a group $G,$ such that $G$ is equal to the $n$-fold product $M\,M^{-1} M\,M^{-1} \dots\,M^{(-1)^{n-1}},$ but not to any proper initial subproduct of this product. We answer his question affirmatively, and prove a more general result on representing a certain sort of relation algebra by subsets of a group. We also sketch several variants of the latter result.