Abstract Algebra - lifting up a factor group

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SUMMARY

The discussion centers on the concept of "lifting up" factor groups in Abstract Algebra, specifically regarding the relationship between a factor group G/N and its corresponding subgroups within G. The key takeaway is that subgroups of G/N correspond to subgroups of G that contain the normal subgroup N. If there exists a subgroup H such that N < H < G, then H is identified as a lift of the factor group H/N. The conversation highlights the importance of understanding isomorphism and homomorphism theorems in this context.

PREREQUISITES
  • Understanding of factor groups and normal subgroups in group theory.
  • Familiarity with isomorphism and homomorphism theorems in Abstract Algebra.
  • Basic knowledge of subgroup properties and their relationships.
  • Experience with group notation and terminology.
NEXT STEPS
  • Study the properties of normal subgroups and their role in factor groups.
  • Learn about the isomorphism theorems in greater detail, particularly in relation to factor groups.
  • Explore examples of lifting subgroups in various group structures.
  • Investigate the implications of subgroup relationships in the context of group homomorphisms.
USEFUL FOR

Students of Abstract Algebra, mathematicians exploring group theory, and educators seeking to clarify the concept of lifting factor groups in their teaching.

nebbish
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Abstract Algebra -- lifting up a factor group

After spending an extended period with my Professor during office hours I must admit I am mystified. He kept on talking about "lifting up" factor groups. I think this has something to do with using a factor group, say G/N, to show that there exists a larger group within G itself somehow related to G/N. It would be nice if the relationship were an isomorphism but I don't think we're guaranteed that. I've been looking over all the isomorphism and homomorphism theorems especially as they relate to factor groups but am still lost. The text never uses the phrase "lifting up" and I see nothing in the text or my notes about how to go from G/N back into G itself in any useful way. Note that there are clearly trivial non-useful ways to go from G/N back into G, I just can't find a useful way.

Any idea what he's talking about?
 
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This is now not homework, and you might get a better discussion inthe algebra thread. All he's using are the isomorphism theorems:

Subgroups of G/N are subgroups of G that contain N. If N<H<G, then H is a lift of H/N.
 

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