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Proof of a proposition about homomorphisms Watch

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    I was reading a proof about a proposition, but I'm not sure what the phrase "\alpha restricts to an isomorphism from \iota(H) to \iota'(H)" really means. I was trying to figure it out from the diagram, but I was a bit confused...

    Thanks in advance
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    (Original post by Artus)
    I was reading a proof about a proposition, but I'm not sure what the phrase "\alpha restricts to an isomorphism from \iota(H) to \iota'(H)" really means. I was trying to figure it out from the diagram, but I was a bit confused...

    Thanks in advance
    In general, if we have a homomorphism \phi: A \rightarrow B, and we have C \leq A, D\leq B, to say that phi restricts to an isomorphism from C to D means that \phi(C)=D (i.e. the image of C under phi is D), and that the restriction \phi\restriction_C : C \rightarrow D is an isomorphism.
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    To add to the above - in this instance, it just means apply the isomorphism to elements of the form: (h,1_K) \in H \rtimes K \rightarrrow. The condition says that the isomorphism on the semidirect product takes such elements to elements of the form (h',1_k) bijectively
 
 
 
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