Answer to Question #105619 in Abstract Algebra for Sourav Mondal

Question

The set of cosets of <(1 2)> in S3 is a group with respect to multiplication of cosets.

Answer 1

Let ,$H=\frac{S_3}{<(1,2)>}=$ ${$ $<(1,2)>,(1,3)<(1,2)>,(2,3)<(1,2)>$ $}$

Where,<(1,2)>= {1,(1,2)} ,

(1,3)$<$ (1,2)$>=$ ${$ (1,3),(1,2,3)$}$ $=(1,2,3)<(1,2)>$

(2,3)$<$ (1,2)$>$ $=$ ${$ (2,3),(1,3,2)$}$ $=(1,3,2)<(1,2)>$ .

Since ,$(1,3)<(1,2)>(2,3)<(1,2)>=(1,3)(2,3)<(1,2)>$

$=(1,3,2)<(1,2)>=(2,3)<(1,2)>$

And $(2,3)<(1,2)>(1,3)<(1,2)>=(2,3)(1,3)<(1,2)>$

$=(1,2,3)<(1,2)>=(1,3)<(1,2)>$

Hence,$H$ is closed.

Clearly,<(1,2)> is the identity element of H and each element is self inverse .

Since $S_3$ Is associative so,$H$ also.

Therefore, H is a group.