Question

Let G = S5 and let H be the subgroup ⟨(123)(45)⟩. (a) Find the number of left cosets of H in G. (b) Which left coset contains the element (13425)? Find all elements of this coset.

Answer 1

The number of left cosets of H in G is 10. The left coset that contains the element (13425) is {(13425), (143)(25), (125)(43), (12453), (15423), (15324), (14235), (13524), (15243), (14532)}.

Step-by-step explanation:

(a) To find the number of left cosets of H in G, we can use the formula: #left cosets = |G|/|H|, where |G| is the order of G and |H| is the order of H. In this case, G = S5, which has 5! = 120 elements. H is the subgroup generated by the permutation (123)(45), which has 3! * 2! = 12 elements. Therefore, the number of left cosets of H in G is 120/12 = 10.

(b) To find which left coset contains the element (13425), we can multiply that element on the right by each element of H and see where it ends up. We have:

(13425)(123)(45) = (143)(25)

(13425)(45)(123) = (125)(43)

(13425)(123)(45)(123) = (12453)

(13425)(45)(123)(45) = (15423)

(13425)(123)(45)(123)(45) = (15324)

(13425)(45)(123)(45)(123) = (14235)

(13425)(123)(45)(123)(45)(123) = (13524)

(13425)(45)(123)(45)(123)(45) = (15243)

(13425)(123)(45)(123)(45)(123)(45) = (14532)

(13425)(45)(123)(45)(123)(45)(123) = (13425)

Thus, the element (13425) is in the left coset {(13425), (143)(25), (125)(43), (12453), (15423), (15324), (14235), (13524), (15243), (14532)}.