Answer to Question #179484 in Algebra for swarnajeet kumar

Question

 a) Using Cayley’s theorem, find the permutation group to which a cyclic group of 

order 12 is isomorphic. (4) 

 b) Let τ be a fixed odd permutation in .

S10 Show that every odd permutation in S is 

10

a product of τ and some permutation in .

A10 (2) 

 c) List two distinct cosets of < r > in ,

D10 where r is a reflection in .

D10 (2) 

 d) Give the smallest n ∈ N for which An is non-abelian. Justify your answer. 

Answer 1

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 d) Give the smallest n ∈ N for which An is non-abelian. Justify your answer. 

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10.04.21

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You

15:24

We get the triangle, let call side AB - c, AC-b, BC-a

1)So,using the law of cosines:

a^2 = b^2 + c^2 - 2 bc\times cos A\\ a^2 = 8^2 + 11^2 - 2 (8)(11)\times cos 60\degree\\ a^2 = 97,\\ a = 9.8 kma 

2

 =b 

2

 +c 

2

 −2bc×cosA

a 

2

 =8 

2

 +11 

2

 −2(8)(11)×cos60°

a 

2

 =97,

a=9.8km

BC=9.8 km

2)

From the law of Sines find the angle C at point C

\frac{9.8}{sin60 \degree} = \frac{11}{sin C}\\ sin C = \frac{sin60 \degree \times 11}{9.8}=0.972\\ C=sin^{-1}(0.972 )= 75.3 \degree 

sin60°

9.8

​

 = 

sinC

11

​

 

sinC= 

9.8

sin60°×11

​

 =0.972

C=sin 

−1

 (0.972)=75.3°

The bearing of B from C is the angle formed by the line joining C and B and rotating about C. By Geometry this angle is

180° - (C + 19°) = 180°-(75.3°+19°)=85.7 \degree°

You

15:27

We get the triangle, let call side AB - c, AC-b, BC-a

A)So,using the law of cosines:

a^2 = b^2 + c^2 - 2 bc\times cos A\\ a^2 = 8^2 + 11^2 - 2 (8)(11)\times cos 60\degree\\ a^2 = 97,\\ a = 9.8 kma 

2

 =b2+c 2 −2bc×cosAa 2=8 2+11 

 −2(8)(11)×cos60°a 2

 =97,

a=9.8km

BC=9.8 km

From the law of Sines find the angle C at point C

\frac{9.8}{sin60 \degree} = \frac{11}{sin C}\\ sin C = \frac{sin60 \degree \times 11}{9.8}=0.972\\ C=sin^{-1}(0.972 )= 75.3 \degree 

sin60°

9.8

​

 = sinC11

B)

 

sinC= 

9.8

sin60°×11

​

 =0.972

C=sin 

−1

 (0.972)=75.3°

C)

The bearing of B from C is the angle formed by the line joining C and B and rotating about C. By Geometry this angle is

180° - (C + 19°) = 180°-(75.3°

+19°)=85.7 \degree°