Problem Solving Set 20

22 July 2012

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

2. If n = 4444^{4444}, a is the sum of all the digits of n, b is the sum of all the digits of a, and c is the sum of all the digits of b, find c.