wu :: forums - Print Page


    
      
        wu :: forums
        (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi)
      

        riddles >> putnam exam (pure math) >> Sequences and Subgroups
        
(Message started by: ecoist on Jun 11^th, 2006, 8:02pm)

Title: Sequences and Subgroups
Post by ecoist on Jun 11^th, 2006, 8:02pm

When Hans Zassenhaus was only 18, he wrote a book on group theory which contains the exercise

Let G be a group of order n. Show that, given any sequence x₁,...,x_n, of elements of G of length n, some consecutive subsequence, x_i,...,x_j, 1<=i<=j<=n, has product x_i...x_j equal to the identity of G.

What about the converse?

Let G be a group containing an n-subset H with the property that, for every sequence of elements of H of length n, some consecutive subsequence has product equal to the identity of G. Must H be a subgroup of G?