Answer to Question #177266 in Abstract Algebra for Abhijeet

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

Let N denote the set of positive integers, and let R denote the relation < in N, i.e. 

( b) E R if a < b. Hence (a, b) E R -1 iff a> b

R 0 R -1 = { (x, y) : x,y E N; 3 b E N s.t. (x, c) E R -1 , (b, y) E R 

= {(x, y) : x,y E N; 3b E N s.t. b < x, b < y} 

(N\{1}) X (N\{1}) = 

{(x,y): x,y E N; x,y 1) 

and 

R -1 OR = { (x, y) x,y E N; 3b E N s.t. (x, E R, (b,y) E R -1 } 

{ (x, y) : x,y E N; 3b E N s.t. b > x, b> y} 

N X N 

Note that R 0 R -1 R -1 o R. 

d) [a] = [b]. If Fain [b] 0, there exists an element x E A with x E [a] n [b]. Hence (a, 

x) 

E 

and (b, x) E R. By symmetry, (x, b) E R and, by transitivity, (a, h) E R. Consequently and is Non abelian