Embedding Group as a Normal Subgroup

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SUMMARY

The discussion centers on embedding a group G into another group H such that G is a normal subgroup of H. The primary method mentioned is the embedding G → G x G', where G' is another group. The participants suggest exploring semidirect products as an alternative approach, particularly noting that the process may be simpler when G is an Abelian group embedded in an Abelian group.

PREREQUISITES
  • Understanding of group theory concepts, specifically normal subgroups.
  • Familiarity with semidirect products in group theory.
  • Knowledge of Abelian groups and their properties.
  • Basic concepts of group embeddings and homomorphisms.
NEXT STEPS
  • Research the properties of normal subgroups in group theory.
  • Study the construction and applications of semidirect products.
  • Explore examples of embedding Abelian groups into larger groups.
  • Investigate the implications of group homomorphisms in the context of normal subgroups.
USEFUL FOR

Mathematicians, particularly those specializing in abstract algebra, group theorists, and students seeking to deepen their understanding of group embeddings and normal subgroups.

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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