Narrow Results

An algorithm for simulation of laser/particle interactions using normalized units related to different laser frequencies was studied. Correct and wrong conversion methods between different normalized units were investigated in simulations of vacuum laser acceleration. The dynamic characteristics obtained with different processing methods differ greatly and some good acceleration results are actually no other than errors stemming from incorrect treatments of the normalized units.

Let p be a prime and let F be a finite field of characteristic p. Let FG denote the group algebra of the finite p-group G over the field F and let $V(FG)$ denote the group of normalized units in FG. The anti-automorphism $g\mapsto g^{-1}$ of G extends linearly to an anti-automorphism $a\mapsto a^{\ast }$ of FG. An element $u\in V(FG)$ is called unitary if $u^{\ast }=u^{-1}$. All unitary elements of $V(FG)$ form a subgroup which is denoted by $V_{\ast }(FG)$. If p is odd, the order of $V_{\ast }(FG)$ is $|F{|}^{(|G|-1)/2}$. However, to compute the order of $V_{\ast }(FG)$ still is open when $p=2$. In this paper, the order of $V_{\ast }(FG)$ is computed when G is a nonabelian $2$-group given by a central extension of the following form:

Let $D_{2n}$ be the dihedral group of order $2n$. In this paper, we provide a description for the group of units $U(\mathbb{Z}{D}_8)$ and $U(\mathbb{Z}{D}_{12})$. For both of these dihedral groups, their corresponding torsion free normal complements are also characterized in their respective groups of normalized units.