We examine the existence of universal elements in classes of infinite abelian groups. The main method is using group invariants which are defined relative to club guessing sequences. We prove, for example: Theorem: For n ≥ 2, there is a purely universal separable p-group in אn if, and only if, 20 ≤ אn. §0 Introduction In this paper “group” will always mean “infinite abelian group”, and “cardina...

‎in [u‎. ‎dempwolff‎, ‎on extensions of elementary abelian groups of order $2^{5}$ by $gl(5,2)$‎, ‎textit{rend‎. ‎sem‎. ‎mat‎. ‎univ‎. ‎padova}‎, ‎textbf{48} (1972)‎, ‎359‎ - ‎364.] dempwolff proved the existence of a group of the‎ ‎form $2^{5}{^{cdot}}gl(5,2)$ (a non split extension of the‎ ‎elementary abelian group $2^{5}$ by the general linear group‎ ‎$gl(5,2)$)‎. ‎this group is the second l...

In this paper, we show that good structured codes over non-Abelian groups do exist. Specifically, we construct codes over the smallest non-Abelian group D6 and show that the performance of these codes is superior to the performance of Abelian group codes of the same alphabet size. This promises the possibility of using non-Abelian codes for multi-terminal settings where the structure of the cod...

let $g$ be a group and $aut(g)$ be the group of automorphisms of‎‎$g$‎. ‎for any natural‎‎number $m$‎, ‎the $m^{th}$-autocommutator subgroup of $g$ is defined‎‎as‎: ‎$$k_{m}(g)=langle[g,alpha_{1},ldots,alpha_{m}] |gin g‎,‎alpha_{1},ldots,alpha_{m}in aut(g)rangle.$$‎‎in this paper‎, ‎we obtain the $m^{th}$-autocommutator subgroup of‎‎all finite abelian groups‎.

a $p$-group $g$ is $p$-central if $g^{p}le z(g)$‎, ‎and $g$ is‎ ‎$p^{2}$-abelian if $(xy)^{p^{2}}=x^{p^{2}}y^{p^{2}}$ for all $x,yin‎ ‎g$‎. ‎we prove that for $g$ a finite $p^{2}$-abelian $p$-central ‎$p$-group‎, ‎excluding certain cases‎, ‎the order of $g$ divides the ‎order of $text{aut}(g)$‎.

We introduce a syntax for quotient types in a predicate logic over a simply type theory. To illustrate its usefulness we construct in purely type theoretic terms (a) the free abelian group on a commutative monoid H, as quotient of H H; a special instance is the construction of Z from N; (b) the quotient poset of a preorder, (c) the abelian quotient of an arbitrary group, and (d) tensor products...