Normal Subgroup Conjugate of H by element

In summary, a normal subgroup conjugate of H by element is a subgroup that is obtained by conjugating the elements of H by a chosen element of the larger group. This concept is important as it helps us understand the structure and properties of a group and allows us to classify groups and determine their isomorphism. To determine if a subgroup is a normal subgroup conjugate of H by element, we need to check if the subgroup is closed under conjugation by the chosen element. A normal subgroup and its conjugate can be different because a normal subgroup is permuted by all elements of the larger group while a conjugate subgroup is only permuted by one chosen element. Normal subgroup conjugation is a specific case of normal subgroups, where a
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Homework Statement


Let H be a subgroup of group G. Then
[itex]H \unlhd G \Leftrightarrow xHx^{-1}=H \forall x\in G [/itex]
[itex] \Leftrightarrow xH=Hx \forall x\in G [/itex]
[itex] \Leftrightarrow xHx^{-1}=Hxx^{-1} \forall x\in G [/itex]
[itex] \Leftrightarrow xHx^{-1}=HxHx^{-1}=H \forall x\in G [/itex]

Homework Equations


xH={xh:h in H}

The Attempt at a Solution


Reviewed the definitions and properties of cosets/normal subgrps, but could not understand the last step. That is how to get from [itex] Hxx^{-1} to HxHx^{-1}[/itex]

EDIT:
Is it simply because [itex] xH=H \Leftrightarrow x \in H [/itex] ?
 
Last edited:
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  • #2

Yes, the last step follows from the fact that if x is an element of H, then xH = H. This can be shown by noting that for any h in H, xh is also in H (since H is a subgroup and x is in H), so xH is a subset of H. Similarly, for any h' in H, x^{-1}h' is also in H, so H is a subset of xH. Therefore, xH = H.
 

FAQ: Normal Subgroup Conjugate of H by element

What is a normal subgroup conjugate of H by element?

A normal subgroup conjugate of H by element is a subgroup that is obtained by conjugating the elements of H by a chosen element of the larger group. In other words, it is a subgroup that is similar to H but has its elements permuted by the chosen element.

Why is the concept of normal subgroup conjugate important?

The concept of normal subgroup conjugate is important because it helps us understand the structure and properties of a group. It also allows us to classify groups and determine their isomorphism.

How do you determine if a subgroup is a normal subgroup conjugate of H by element?

To determine if a subgroup is a normal subgroup conjugate of H by element, we need to check if the subgroup is closed under conjugation by the chosen element. This means that if we conjugate any element of the subgroup by the chosen element, the result should still be in the subgroup.

Can a normal subgroup and its conjugate be different?

Yes, a normal subgroup and its conjugate can be different. This is because the elements of a normal subgroup are permuted by all elements of the larger group, while the elements of a conjugate subgroup are permuted by only one chosen element.

How does normal subgroup conjugation relate to normal subgroups?

Normal subgroup conjugation is a specific case of normal subgroups. A normal subgroup is a subgroup that is invariant under conjugation by all elements of the larger group, while a normal subgroup conjugate is only invariant under conjugation by a chosen element. In other words, all normal subgroup conjugates are normal subgroups, but not all normal subgroups are normal subgroup conjugates.

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