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

Let D be the diagonal subset D={(x,x)∣x∈S3} 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,y∈S3. Then (x,x)⋅(y,y)−1=(x,x)(y−1,y−1)=(xy−1,xy−1)∈D. Therefore, D is subgroup of S3×S3. Let g,h∈S3×S3, such that g=((1,2),(1,3)) and h=(1212)∈D Now g−1=((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.

Expert answers to "Let D be the diagonal subset D={(x,x)∣x∈S3} of..."

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ankarskogC

Answered question

2021-01-25

Let D be the diagonal subset $D = {(x, x) ∣ x \in S_{3}}$ of the direct product $S_{3} \times S_{3}$ . Prove that D is a subgroup of $S_{3} \times S_{3}$ but not a normal subgroup.

Answer & Explanation

Asma Vang

Skilled2021-01-26Added 93 answers

Let D be the diagonal subset $D = {(x, x) ∣ x \in S 3}$ of the direct product $S 3 \times S 3$ , where
$S 3 = {e, (1, 2), (2, 3), (1, 3), (1, 2, 3), (1, 3, 2)}$
To show that D is a subgroup of $S 3 \times S 3$ . We know that $(e, e) \in D$ so, D is non-empty. Now suppose $(x, x), (y, y) \in D$ for some $x, y \in S 3$ . Then
$(x, x) \cdot {(y, y)}^{- 1} = (x, x) (y^{- 1}, y^{- 1}) = (x y^{- 1}, x y^{- 1}) \in D .$
Therefore, D is subgroup of $S 3 \times S 3$ . Let $g, h \in S 3 \times S 3$ , such that $g = ((1, 2), (1, 3))$ and $h = (\begin{array}{cc} 1 & 2 \\ 1 & 2 \end{array}) \in D$
Now $g^{-} 1 = ((1, 2), (1, 3)) ((1, 2) (1, 2)) ((1, 2) (1, 3))^{-} 1$

$= ((1, 2) (1, 3)) ((1, 2) (1, 2)) ((1, 2)^{-} 1 (1, 3)^{-} 1)$

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

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

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

$= ((1, 2) (2, 3)) \in D .$
Therefore, D is not normal subgroup of $S 3 \times S 3$ .

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