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I don't know the answer. However, this is what I would try. $\alpha$ is constrained to have as a subword $ac^ma$, for nonzero $m$ and where I abuse notation and use $a$ to represent itself or its in …

If $k$ is allowed to be much, much larger than $n$, then no. A consequence of the assumption is that $a$ and $b$ each have fixed points. Let's take a toy example and see for what $n$ the example wor …

It strikes me that you are going to need to remember either the index i or the value pi(i), so barring "magic memory", you are going to need log n bits as a strict minimum. For small n, you may as we …

I want to share a partial answer to question 1), and raise a few more questions. I found what I think is a neat and likely unoriginal bijection; I'm hoping the combinatorialists can provide a referen …

A common technique of study is to focus on a subset of the properties or a portion of the object under consideration; by understanding a small piece, one may be able to say something about the whole, …

It sounds like richness shares with beauty: the beholder's eye determines what is rich. While I will not attempt to give a quantitative answer, I think the following points can be considered. The o …

To expand on an earlier answer, I will plagiarize an answer provided by Derek Holt in a sci.math thread: A "standard" example of that is the Baumslag-Solitar group $$G = \langle x,y \mid y x y^{-1} = …

I am hoping to learn enough about additive permutations to help with a number theory problem. These permutations also have connections to difference sets, orthomorphisms, transversals, and other stru …

I imagine that definitions of $Mod, Th, Var$ and so on have not changed since I saw them decades ago. The trivial one element algebra in any finite type (and likely any infinite type) is an easy exam …