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IMC 2010 Problem 3, Day 2

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July 27, 2010
group theoryabstract algebrainductionstrong inductionsuperior algebrasuperior algebra unsolved

Problem Statement

Denote by SnS_n the group of permutations of the sequence (1,2,…,n).(1,2,\dots,n). Suppose that GG is a subgroup of Sn,S_n, such that for every π∈G∖{e}\pi\in G\setminus\{e\} there exists a unique k∈{1,2,…,n}k\in \{1,2,\dots,n\} for which π(k)=k.\pi(k)=k. (Here ee is the unit element of the group Sn.S_n.) Show that this kk is the same for all π∈G∖{e}.\pi \in G\setminus \{e\}.
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