(a) An unordered pair fm; ng with 1 ≤ m ≠ n ≤ 6 is called a duad. List the 15 duads.
(b) There are 15 ways to partition {1, ......, 6 } into 3 duads, such as { {1; 2}, {3, 4},{5, 6}}. Such a set of 3 duads is called a syntheme. List the 15 synthemes.
(c) A pentad P is a set of 5 distinct synthemes, such that each duad occurs in Q exactly once. Here is one pentad: P_{1} = {12, 34, 56}; {13, 25, 46}, {14, 26, 35}, {15, 24, 36}, {16, 23, 45}.
Show that there are exactly 6 dierent pentads.
(d) Let σ ∈ S_{6}. Throughout each pentad, replace i by (i). Show that this gives an action of S6 on the set of pentads.
(e) The action above induces a homomorphism : S_{6} -> S_{6}. Show that Ψ is an automorphism. Hint: show it is injective.
(f) Show that is not given by conjugation. Hint: show it does not preserve cycle structure.