Embedding Group as a Normal Subgroup

Thread starter WWGD
Start date Apr 25, 2014
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In summary, a normal subgroup is a subgroup of a group that is invariant under conjugation by elements of the larger group. It is embedded by identifying a subgroup within a larger group that is invariant under conjugation. This allows for the construction of quotient groups, which are essential in understanding the structure and properties of a larger group. The normal subgroup is related to the quotient group by forming the elements of the quotient group through cosets and defining the group operation through coset multiplication. Not all subgroups can be embedded as normal subgroups, only those that are invariant under conjugation, also known as normalizers.

Apr 25, 2014

WWGD

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Hi, let G be any group . Is there a way of embedding G in some other
group H so that G is normal in H, _other_ than by using the embedding:

G -->G x G' , for some group G'?

I assume this is easier if G is Abelian and is embedded in an
Abelian group. Is there a way of doing this in general?

Thanks.

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Apr 25, 2014

micromass

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1. What is a normal subgroup?

A normal subgroup is a subgroup of a group that is invariant under conjugation by elements of the larger group. This means that for any element in the normal subgroup and any element in the larger group, their product and inverse will also be in the normal subgroup.

2. How is a group embedded as a normal subgroup?

A group is embedded as a normal subgroup by identifying a subgroup within a larger group that is invariant under conjugation by elements of the larger group. This subgroup then becomes the normal subgroup and can be used to define a quotient group.

3. What is the significance of embedding a group as a normal subgroup?

Embedding a group as a normal subgroup allows for the construction of quotient groups, which are essential in understanding the structure and properties of a larger group. It also helps in proving theorems and solving problems related to the group.

4. How is the normal subgroup related to the quotient group?

The normal subgroup is used to construct the quotient group by taking the cosets of the normal subgroup in the larger group. These cosets form the elements of the quotient group, and the group operation is defined by the coset multiplication.

5. Can any subgroup be embedded as a normal subgroup?

No, not all subgroups can be embedded as normal subgroups. Only subgroups that are invariant under conjugation by elements of the larger group can be considered as normal subgroups. These subgroups are also known as normalizers.