Locally parabolic subgroups in Coxeter groups of arbitrary ranks

Despite the significance of the notion of parabolic closures in Coxeter groups of finite ranks, the parabolic closure is not guaranteed to exist as a parabolic subgroup in a general case. In this paper, first we give a concrete example to clarify that the parabolic closure of even an irreducible reflection subgroup of countable rank does not necessarily exist as a parabolic subgroup. Then we propose a generalized notion of "locally parabolic closure" by introducing a notion of "locally parabolic subgroups", which involves parabolic ones as a special case, and prove that the locally parabolic closure always exists as a locally parabolic subgroup. It is a subgroup of parabolic closure, and we give another example to show that the inclusion may be strict in general. Our result suggests that locally parabolic closure has more natural properties and provides more information than parabolic closure. We also give a result on maximal locally finite, locally parabolic subgroups in Coxeter groups, which generalizes a similar well-known fact on maximal finite parabolic subgroups.Comment: 7 pages; (v2) 11 pages, examples added, main theorem slightly updated (v3) references updated, minor changes performed, to appear in Journal of Algebr

On Compression Functions over Small Groups with Applications to Cryptography

In the area of cryptography, fully homomorphic encryption (FHE) enables any entity to perform arbitrary computation on encrypted data without decrypting the ciphertexts. An ongoing group-theoretic approach to construct FHE schemes uses a certain "compression" function $F(x)$ implemented by group operators on a given finite group $G$ (i.e., it is given by a sequence of elements of $G$ and variable $x$), which satisfies that $F(1) = 1$ and $F(\sigma) = F(\sigma^2) = \sigma$ where $\sigma \in G$ is some element of order three. The previous work gave an example of such $F$ over $G = S_5$ by just a heuristic approach. In this paper, we systematically study the possibilities of such $F$. We construct a shortest possible $F$ over smaller group $G = A_5$, and prove that no such $F$ exists over other groups $G$ of order up to $60 = |A_5|$.Comment: 10 page

A Simple and Elementary Proof of Zorn's Lemma

We give a new simple proof of Zorn's Lemma (from the Axiom of Choice), which is elementary and does not rely on advanced knowledge in set theory (such as transfinite recursion) nor in ordered sets (such as well-ordered sets) beyond the statement of Zorn's Lemma itself.Comment: 2 page