Question

what is the order of the group A6 intersection D6?

Answer 1

We know that, for every subgroup H of Sn, either H is contained in An or exactly half of the elements of H are in An.

D6 is a subgroup of S6 in a natural way. We also have that the number of elements in D6 =12.

Note that the element (1 2 3 4 5 6), which corresponds to 60° rotation, is in D6. However, being a cycle of length 6 (even length) , it is an odd permutation, as

(1 2 3 4 5 6)= (1 6) (1 5) (1 4) (1 3) (1 2)

which is a product of odd number of transpositions.

Thus, exactly half of the elements of D6 are in A6, i.e,

the order of the group A6 intersection D6 = 12/2 = 6.