1. Is the map Z ⇥ Z ! S3 given by (i, j) 7! (12)i (123)j a homomorphism? Prove your answer. 2. Find all possible orders in A5. (Do not list all the elements, just think of all the possible structures of a permutation in A5 as a product of disjoint cycles). 3. Prove, using induction on n, that every element in Sn is a product of transpositions. 4. Prove that every permutation in An can be written as a product of 3-cycles.

1. Is the map Z ⇥ Z ! S3 given by (i, j) 7! (12)i
(123)j a homomorphism? Prove your
answer.
2. Find all possible orders in A5. (Do not list all the elements, just think of all the possible
structures of a permutation in A5 as a product of disjoint cycles).
3. Prove, using induction on n, that every element in Sn is a product of transpositions.
4. Prove that every permutation in An can be written as a product of 3-cycles.
5. Suppose G is a group, a 2 G and H is a subgroup of G.
(a) Show that if aH = H then a 2 H.
(b) Show that if a 2 H, then aH = H. (Hint: Assume a belongs to H. Then prove
double inclusion: aH is a subset of H, and viceversa.)
(Together, they imply that aH = H if and only if a 2 H. )
6. Suppose G is a group, a 2 G and H is a subgroup of G.
(a) Show that aH and H have the same cardinality by exhibiting a bijection between
the two sets. (Similarly, one can show that Ha and H have the same cardinality, so
aH and Ha always have the same cardinality.)
(b) Show, by means of an example, that aH is not necessarily equal to Ha. (Hint: it
has to be a non-commutative group)
7. Suppose G is a group, a 2 G and H is a subgroup of G. Show that aH = Ha if and only
if aHa1 = H.
8. (a) Recall that we can view D2n as a subgroup of Sn. Find the partition of S4 into left
cosets of D8.
(b) Let H = SL2(R) denote the subgroup of GL(2, R) consisting of all matrices with
determinant 1. Describe the partition of GL(2, R) into left cosets of H.