Course 111: Algebra, 17th November 2006 {e, a, a } and a

Course 111: Algebra, 17th November 2006
1. Suppose G is a cyclic group of order 4, G = {e, a, a2 , a3 } and a4 = e.
Find the order of each element of G. Find all the subgroups of G.
List the distinct left cosets of the subgroup H = {e, a2 } in G.
2. Prove that for H and K both subgroups of a group G. The product,
HK is itself a subgroup of G iff HK = KH.
Note: HK = {x ∈ G|x = hk, h ∈ H, k ∈ K}
3. Recall the properties of cosets listed in your notes. From this list prove
the following:
For a subgroup H of a group G,
• x ∈ xH for all x ∈ G.
• if x and y ∈ G and if y = xa for some a ∈ H then xH = yH.
• each left coset of H in G has the same number of elements as H.