Course 111: Algebra, 22nd November 2006

Course 111: Algebra, 22nd November 2006
To be handed in at tutorials on Nov 27th and 28th.
1. Consider the set of elements
G = {0, 1, 2, 3, 4, 5}
which forms a cyclic group under addition modulo 6.
List the proper subgroups of G and the generators of the group.
(Recall that {G} and {e}, where e = 0 here, are improper subgroups)
Determine the left and right cosets of H = {0, 3} in G and construct
the Cayley table of the quotient group G|H.
2. 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.