Tutorial K Solutions PDF

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Tutorial K Solutions Tuesday, March 31, 2009 Problem 1. Let α= Compute the following: of disjoint cycles. ! 1 2 3 Asnwer: β ◦ α = 2 1 3 ! 3 4 1 α−1 = 1 2 3 ! 4 3 1 −1 β = 1 2 3

!

1 2 3 4 3 4 1 2

"

and β =

!

1 2 3 4 4 3 1 2

"

β ◦ α, α ◦ β, α−1 , β −1 , (α ◦ β)−1 ; and write them as products " 4 = (1 4 " ! 1 2 = 3 4 " ! 2 1 = 4 3

2); α ◦ β =

!

1 2 3 4 1 2 4 3

"

2 3 4 4 1 2

"

= (1 3)(2 4)

2 3 4 4 2 1

"

= (1

3

2

= (1 2)

4)

(α ◦ β )−1 = β −1 ◦ α−1 = ( 3 4 ) . Problem!2. Write each of the " following as a single cycle or a product of disjoint cycles. 1 2 3 4 5 6 = (1 3 6)(2 5) Answer: (a) 3 5 6 4 2 1 (b) ( 1 (c)

2)(1

3)(1

( 1 3 ) −1 ( 2

(d) ( 1

4

5)(1

4) = (1

4)(2

3

2

5)(1

3

2

3

4)

5 ) −1 = ( 1 3) = (1

3)(2

4)(2

4

5

3

5

3)

2).

Problem 3. The pentagonal prism has 2 pentagonal faces, 5 square face, and ten vertices. Label these vertices from 1 to 10 and express the rotational symmetries of the pentagonal prism as permuations on these ten numbers.

1 2 5 6 7

3

4 0

8 9 over→ 1

1 2 5 6 7

3

4 0

8 9 Note: For convenience, I have changed “10” to “0”. A rotation through 2π about an axis passing through the mid-points of the two 5 pentagons is ( 1 2 3 4 5 ) ( 6 7 8 9 0 ). Rrotation through twice this angle about the same axis is ( 1 3 5 2 4 ) ( 6 8 0 7 9 ); through three times this angle: ( 1 4 2 5 3 ) ( 6 9 7 0 8 ) ; and through four times this angle ( 1 5 4 3 2 ) ( 6 0 9 8 7 ) . Rotation through five times this angle gives the identity (1). We can rotate the prism through π about any line joining the midpoint of a vertical side, say the side 16 to the mid-point of the opposite square face, the face 3894 in sthis example. This permutation is: ( 1 6 ) ( 2 0 ) ( 3 9 ) ( 4 8 ) ( 5 7 ) . Similarly, there are four more such rotations through π about lines joining the midpoint of a vertical edge to the midpoint of the opposite square face. Altogether there are 10 rotations.

2...