Undergrad Group extensions (question about definitions)

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The discussion centers on clarifying the definition of group extensions, specifically the notation ##\mathcal{E}(G,N)##, and whether all extensions are isomorphic or diffeomorphic. There is a request for an explanation of the pullback ##u^*## in relation to the mapping ##u:G' \rightarrow G##. The user expresses a need for foundational resources to better understand the topic. Overall, the conversation aims to deepen comprehension of group extensions and related concepts.
Korybut
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Definition question
Hello!

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

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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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