Write down all the permutations of X as
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Write down all the permutations of X as

2. Let X = {1, 2, 3, 4}.

(a) Write down all the permutations of X as (2 × 4)-matrices.

(b) Which of these permutations are symmetries of the square with vertices 1, 2, 3 and 4?

For each one that is a symmetry, give it the name ρi (for a suitable choice of i) if it is a rotation and σj (for a suitable choice of j) if it is a reflection.

(c) Using your answers to part (b), construct the Cayley table for the dihedral group D4.

(d) Find all subgroups of D4.

Hint
MathematicsPermutations of a set refer to an arrangement of members of the set into a sequence or linear order. Permutations are important because they contribute to counting problems as well as other disciplines such as mathematics where the determinant of a sum is defined using permutations....
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