2021/03/30 by Uwe Schauz, Schauz, Uwe
Computer Science · Mathematics · #13F20 #20C05 #20K01 #41A05 #Advanced Topics in Algebra #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #math.GR #msc:13F20 #msc:20C05 #msc:20K01 #msc:41A05
paper · pdf · doi:10.48550/arxiv.2103.16467
18 pages
openalex publication_date 2021/03/30 · arxiv created 2021/06/25 · arxiv updated 2021/06/28 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We consider maps between commutative groups and their functional degrees. These degrees are defined based on a simple idea -- the functional degree should decrease if a discrete derivative is taken. We show that the maps of finite functional degree are precisely the maps that can be written as polyfracts, as polynomials in several variables but with binomial functions in the place of powers. Moreover, the degree of a polyfract coincides with its functional degree. We use this to determine the largest possible finite functional degree that the maps between two given finite commutative groups can have. This also yields a solution to Aichinger and Moosbauer's problem of finding the nilpotency degree of the augmentation ideal of the group ring Zpβ[Zpα1× Zpα2×…× Zpαn]. Some generalizations and simplifications of proofs to underlying facts are presented, too.