(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: P₁ = {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 ∈ S₆. 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  : S₆ -> S₆. 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.

 

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