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Inequalities with |H|

Source: Romanian MO 2010 Grade 12

August 6, 2012
inequalitiesgroup theoryabstract algebrasuperior algebrasuperior algebra unsolved

Problem Statement

Let GG be a finite group of order nn. Define the set H={x:x∈G and x2=e},H=\{x:x\in G\text{ and }x^2=e\}, where ee is the neutral element of GG. Let p=∣H∣p=|H| be the cardinality of HH. Prove that a) ∣H∩xH∣≥2p−n|H\cap xH|\ge 2p-n, for any x∈Gx\in G, where xH={xh:h∈H}xH=\{xh:h\in H\}. b) If p>3n4p>\frac{3n}{4}, then GG is commutative. c) If n2<p≤3n4\frac{n}{2}<p\le\frac{3n}{4}, then GG is non-commutative.
Marian Andronache
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