math assignment

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hw06.pdf

HOMEWORK 6

Due Thursday, May 25, at the beginning of discussion

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 aHa

�1 = 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.

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