Let D be the diagonal subset D = {(x, x)|x ∈ S_3} of the direct product S_3 × S_3. Prove that D is a subgroup of S_3 × S_3 but not a normal subgroup.

ankarskogC

ankarskogC

Answered question

2021-01-25

Let D be the diagonal subset D={(x,x)xS3} of the direct product S3×S3. Prove that D is a subgroup of S3×S3 but not a normal subgroup.

Answer & Explanation

Asma Vang

Asma Vang

Skilled2021-01-26Added 93 answers

Let D be the diagonal subset D={(x,x)xS3} of the direct product S3×S3, where
S3={e,(1,2),(2,3),(1,3),(1,2,3),(1,3,2)}
To show that D is a subgroup of S3×S3. We know that (e,e)D so, D is non-empty. Now suppose (x,x),(y,y)D for some x,yS3. Then
(x,x)(y,y)1=(x,x)(y1,y1)=(xy1,xy1)D.
Therefore, D is subgroup of S3×S3. Let g,hS3×S3, such that g=((1,2),(1,3)) and h=(1212)D
Now g1=((1,2),(1,3))((1,2)(1,2))((1,2)(1,3))1

=((1,2)(1,3))((1,2)(1,2))((1,2)1(1,3)1)

=((1,2)(1,3))((1,2)(1,2))((1,2)(1,3))

=((1,2)(1,2)(1,2)(1,3)(1,2)(1,3))

=((1,2),(1,3).(1,3,2))

=((1,2)(2,3))D.
Therefore, D is not normal subgroup of S3×S3.

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