Around Podewski's conjecture

A long-standing conjecture of Podewski states that every minimal field is algebraically closed. It was proved by Wagner for fields of positive characteristic, but it remains wide open in the zero-characteristic case. We reduce Podewski's conjecture to the case of fields having a definable (in the pure field structure), well partial order with an infinite chain, and we conjecture that such fields do not exist. Then we support this conjecture by showing that there is no minimal field interpreting a linear order in a specific way; in our terminology, there is no almost linear, minimal field. On the other hand, we give an example of an almost linear, minimal group $(M,<,+,0)$ of exponent 2, and we show that each almost linear, minimal group is elementary abelian of prime exponent. On the other hand, we give an example of an almost linear, minimal group $(M,<,+,0)$ of exponent 2, and we show that each almost linear, minimal group is torsion.Comment: 16 page

Left-ordered inp-minimal groups

We prove that any left-ordered inp-minimal group is abelian, and we provide an example of a non-abelian left-ordered group of dp-rank 2

Minimal group codes over alternating groups

In this work we show that every minimal code in a semisimple group algebra $\mathbb{F}_qG$ is essential if $G$ is a simple group. Since the alternating group $A_n$ is simple if $n=3$ or $n\geq 5$, we present some examples of minimal codes in $\mathbb{F}_qA_n$. For this purpose, if $char(\mathbb{F}_q)> n$, we present the Wedderburn-Artin decomposition of $\mathbb{F}_qS_n$ and $\mathbb{F}_qA_n$ and explicit some of the centrally primitive idempotents of $\mathbb{F}_qS_n$ and $\mathbb{F}_qA_n$.Comment: 16 page

Locally compact groups and locally minimal group topologies

Minimal groups are Hausdorff topological groups G satisfying the open mapping theorem with respect to continuous isomorphisms, i.e., every continuous iso- morphism G \u2192 H, with H a Hausdorff topological group, is a topological isomorphism. A topological group (G,\u3c4) is called locally minimal if there exists a neighbourhood V of the identity such that for every Hausdorff group topology \u3c3 64\u3c4 with V 08\u3c3 one has \u3c3 = \u3c4. Minimal groups, as well as locally compact groups, are locally minimal. According to a well known theorem of Prodanov, every subgroup of an infinite compact abelian group K is minimal if and only if K is isomorphic to the group Zp of p-adic integers for some prime p. We find a remarkable connection of local minimality to Lie groups and p-adic numbers by means of the following results extending Prodanov\u2019s theorem: every subgroup of a locally compact abelian group K is locally minimal if and only if either K is a Lie group, or K has an open subgroup isomorphic to Zp for some prime p. In the nonabelian case we prove that all subgroups of a connected locally compact group are locally minimal if and only if K is a Lie group, resolving Problem 7.49 from Dikranjan and Megrelishvili (2014) in the positive