Group extensions (question about definitions)

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SUMMARY

The discussion centers on the definition and properties of group extensions, specifically the notation ##\mathcal{E}(G,N)##. The user seeks clarification on whether all extensions are isomorphic or diffeomorphic and requests an explanation of the pullback ##u^*## in the context of a morphism ##u:G' \rightarrow G##. Additionally, the user is looking for beginner-friendly resources to better understand these concepts.

PREREQUISITES
  • Understanding of group theory and its terminology
  • Familiarity with the concept of group extensions
  • Basic knowledge of morphisms in algebraic structures
  • Introduction to differential geometry for diffeomorphic concepts
NEXT STEPS
  • Research the properties of group extensions in algebraic topology
  • Study the concept of pullbacks in category theory
  • Explore introductory texts on group theory and extensions
  • Learn about diffeomorphisms and their applications in differential geometry
USEFUL FOR

Mathematicians, students of algebra, and anyone interested in the foundational concepts of group theory and extensions.

Korybut
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Definition question
Hello!

I would like to be sure about my understanding of the definition provided in screenshot below

?hash=9ed08cdafeb4fe3619693b2d5cc5441b.jpg


1. What is this ##\mathcal{E}(G,N)##? I know that not all extension are isomorphic so I wonder What are the elements of ##\mathcal{E}## groups? Or maybe all Es diffeomorphic to each other. Don't know...

2. Please explain how ##u^*## is the pullback of ##u:G^\prime \rightarrow G##

3. Very new to the subject so I would be grateful if someone recommends nice manual explainig basics.

Many thanks in advance
 

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