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This paper describes in basic terms what a Thin Group is, as well as its uses in various subjects.
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A p-group is called powerful if every commutator is a product of pth powers when p is odd and a product of fourth powers when p=2. In the group algebra of a group G of p-power order over a finite field of characteristic p, the group of normalized units is always a p-group. We prove that it is never powerful except, of course, when G is abelian.
In this article, we investigate the existence and schurity problem of association schemes whose thin residues are isomorphic to an elementary abelian $p$-group of rank $2$.
Left braces, introduced by Rump, have turned out to provide an important tool in the study of set theoretic solutions of the quantum Yang-Baxter equation. In particular, they have allowed to construct several new families of solutions. A left brace $(B,+,cdot )$ is a structure determined by two group structures on a set $B$: an abelian group $(B,+)$ and a group $(B,cdot)$, satisfying certain compatibility conditions. The main result of this paper shows that every finite abelian group $A$ is a subgroup of the additive group of a finite simple left brace $B$ with metabelian multiplicative group with abelian Sylow subgroups. This result complements earlier unexpected results of the authors on an abundance of finite simple left braces.
General acceptance of a mathematical proposition $P$ as a theorem requires convincing evidence that a proof of $P$ exists. But what constitutes convincing evidence? I will argue that, given the types of evidence that are currently accepted as convincing, it is inconsistent to deny similar acceptance to the evidence provided for the existence of proofs by certain randomized computations.
Quantum simulators are devices that actively use quantum effects to answer questions about model systems and, through them, real systems. Here we expand on this definition by answering several fundamental questions about the nature and use of quantum simulators. Our answers address two important areas. First, the difference between an operation termed simulation and another termed computation. This distinction is related to the purpose of an operation, as well as our confidence in and expectation of its accuracy. Second, the threshold between quantum and classical simulations. Throughout, we provide a perspective on the achievements and directions of the field of quantum simulation.
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