MathDB
Romania NMO 2022 Grade 12 P4

Source: Romania National Olympiad 2022

April 20, 2022
Ring Theorygroup theoryromania

Problem Statement

Let (R,+,⋅)(R,+,\cdot) be a ring with center Z={a∈R:ar=ra,∀r∈R}Z=\{a\in\mathbb{R}:ar=ra,\forall r\in\mathbb{R}\} with the property that the group U=U(R)U=U(R) of its invertible elements is finite. Given that GG is the group of automorphisms of the additive group (R,+),(R,+), prove that ∣G∣≥∣U∣2∣Z∩U∣.|G|\geq\frac{|U|^2}{|Z\cap U|}.Dragoș Crișan
Back to ProblemsView on AoPS