Solve Group Theory Problem: Proving HuX is G

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SUMMARY

The discussion focuses on proving that the subgroup generated by the union of a normal subgroup H and a subset X of group G, denoted as HuX, is equal to G. The problem involves the natural map v from G to the quotient group G/H and requires demonstrating that if x is an element of G, then x belongs to the generated subgroup . The user explores various approaches, including direct proof and contradiction, to establish the relationship between HuX and G.

PREREQUISITES
  • Understanding of group theory concepts, specifically normal subgroups and quotient groups.
  • Familiarity with the notation and properties of generated subgroups.
  • Knowledge of the natural map in the context of group homomorphisms.
  • Experience with proof techniques, including direct proof and proof by contradiction.
NEXT STEPS
  • Study the properties of normal subgroups in group theory.
  • Learn about the structure and significance of quotient groups in abstract algebra.
  • Explore the concept of generated subgroups and their applications in group theory.
  • Review proof techniques in mathematics, focusing on direct proofs and contradictions.
USEFUL FOR

Students studying abstract algebra, particularly those preparing for exams in group theory, as well as educators and mathematicians interested in advanced group properties and proofs.

JasonRox
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I'm stuck on this one...

I'm studying for my midterm so I'm solving problems for practice. Here's one of them...

Let H be a normal subgroup in G, and let v be the natural map from G to G/H, and let X be a subset of G such that the subgroup generated by v(X) is G/H. Prove that the subgroup generated by H union X (HuX) is G.

I'm trying to do this directly with showing if x is in G, then x is in <HuX> (generated subgroup). I tried doing contradiction too, by assuming <HuX> is some proper subgroup A of G and not G itself.

I'm going to spend more time thinking about this. I'll be back in like 2 hours since I have a meeting, which I'll spend a minute here or there thinking about it.
 
Physics news on Phys.org
What's the pullback of <v(X)>?
 
Good thought!

Not sure where it will lead me, but I'll think about that too.
 

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