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Math  /  Algebra

Question(i) [3pts] x−1y−1xy∈Hx^{-1} y^{-1} x y \in H for all x,y∈Gx, y \in G then III I is normal subgroup of G

Studdy Solution

STEP 1

1. G G is a group.
2. H H is a subgroup of G G .
3. We need to show that H H is a normal subgroup of G G .

STEP 2

1. Understand the definition of a normal subgroup.
2. Use the given condition x−1y−1xy∈H x^{-1} y^{-1} x y \in H to demonstrate that H H is normal.

STEP 3

Recall the definition of a normal subgroup. A subgroup H H of G G is normal if for every element g∈G g \in G and h∈H h \in H , the element g−1hg g^{-1} h g is also in H H .

STEP 4

Use the given condition x−1y−1xy∈H x^{-1} y^{-1} x y \in H for all x,y∈G x, y \in G .
Choose y=h∈H y = h \in H and x=g∈G x = g \in G . Then, by the given condition:
g−1h−1gh∈H g^{-1} h^{-1} g h \in H

STEP 5

Rearrange the expression g−1h−1gh∈H g^{-1} h^{-1} g h \in H to show that g−1hg∈H g^{-1} h g \in H .
Since h−1∈H h^{-1} \in H implies h∈H h \in H (because H H is a subgroup and closed under inverses), we have:
g−1hg∈H g^{-1} h g \in H
This satisfies the condition for H H to be a normal subgroup of G G .

STEP 6

Conclude that since g−1hg∈H g^{-1} h g \in H for all g∈G g \in G and h∈H h \in H , H H is indeed a normal subgroup of G G .

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