MathDB
Subgroup of all elements of prime order and its ordinal, under some conditions

Source: Romania National Olympiad 2014, Grade XII, Problem 4

March 3, 2019
number theoryprime numbersgroup theoryabstract algebra

Problem Statement

Let be a finite group G G that has an element a≠1 a\neq 1 for which exists a prime number p p such that x1+p=a−1xa, x^{1+p}=a^{-1}xa, for all x∈G. x\in G.
a) Prove that the order of G G is a power of p. p. b) Show that H:={x∈G∣ord(x)=p}≤G H:=\{x\in G|\text{ord} (x)=p\}\le G and ord2(H)>ord(G). \text{ord}^2(H)>\text{ord}(G).
Back to ProblemsView on AoPS