Finite groups with trivial class groups

Finite groups with three relative commutativity degrees

‎‎For a finite group $G$ and a subgroup $H$ of $G$‎, ‎the relative commutativity degree of $H$ in $G$‎, ‎denoted by $d(H,G)$‎, ‎is the probability that an element of $H$ commutes with an element of $G$‎. ‎Let $mathcal{D}(G)={d(H,G):Hleq G}$ be the set of all relative commutativity degrees of subgroups of $G$‎. ‎It is shown that a finite group $G$ admits three relative commutativity degrees if a...

Finite Groups with p-Sylow Coverings

On Subsets of Finite Abelian Groups without Non-trivial Solutions

Let D(G) be the maximal cardinality of a set A ⊆ G that contains no non-trivial solution to x1 + · · · + xs − sxs+1 = 0 with xi ∈ A (1 ≤ i ≤ s + 1). Let d(n) = sup rk(H)≥n D(H) |H| , where rk(H) is the rank of H. We prove that for any n ∈ N, d(n) ≤ C ns−2 , where C is a fixed constant depending only on s.

Groups and spaces with all localizations trivial

The genus of a finitely generated nilpotent group G is defined as the set of isomorphism classes of finitely generated nilpotent groups K such that the p-localizations Kp, Gp are isomorphic for all primes p [19]. This notion turns out to be particularly relevant in the study of non-cancellation phenomena in group theory and homotopy theory. In the above definition, the restriction of finite gen...

Math 5520 Class Notes – Finite Abelian Groups

The main goal in this section is to prove the Fundamental Theorem of Abelian Groups, which roughly speaking says that every finite, abelian group is isomorphic to a product of cyclic groups. In other words, if A is a finite, abelian group, then there are positive integers, n1, . . . , nr, so that A ∼= Z/n1Z× · · · × Z/nrZ. It’s possible to have Z/n1Z × · · · × Z/nrZ ∼= Z/m1Z × · · · × Z/msZ whe...