Subgroups and normal subgroups

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SUMMARY

In the discussion, it is established that if G is a group, N is a normal subgroup of G, and H is a subgroup of G with finite quotients G/N and H such that gcd(|G/N|, |H|) = 1, then H must be a subgroup of N. The participants suggest utilizing Lagrange's Theorem and the action of G on the left cosets of G/N by conjugation, as well as the first isomorphism theorem, to explore the relationship between H and N. The focus shifts to analyzing the structure of G/N to derive insights about H.

PREREQUISITES
  • Understanding of group theory concepts, specifically normal subgroups and finite groups.
  • Familiarity with Lagrange's Theorem and its implications for subgroup orders.
  • Knowledge of group actions and how they relate to cosets.
  • Comprehension of the first isomorphism theorem in group theory.
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  • Study the implications of Lagrange's Theorem in finite groups.
  • Learn about group actions and their applications in subgroup analysis.
  • Research the first isomorphism theorem and its role in group homomorphisms.
  • Examine examples of normal subgroups and their properties in various group structures.
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This discussion is beneficial for mathematicians, particularly those specializing in abstract algebra, group theorists, and students seeking to deepen their understanding of subgroup relationships and normal subgroups in finite groups.

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Let G be a group, let N be normal in G and let H be a subgroup of G. Assume that G/N and H are finite and that gcd(|G/N|,|H|)=1. Prove that H is a subgroup of N.

I was thinking about using Lagrange Theorem. and maybe using the fact that G may act on the set of left cosets (G/N) by conjugation.
and the find the kernel of that action and then maybe use the first isomorphism theorem.

But I don't get very far with that.
 
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H also acts on G/N by left multiplication. Furthermore, every element of H is in one of the the left cosets of N.
 
Maybe looking at G isn't the right way to go. What can you learn about H by studying the group G/N? What about N?
 

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