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For the word "induce", I often see it used in the context of mapping, as in the included example. Can someone tell me what this word mean in a mathematical context and how it is used please.

Example:

Let $H$ and $K$ be normal subgroups of the group $G$. Verify that if the natural mapping $\text{nat}_{K}:G\rightarrow {G/K},$ $\textbf{induces}$ an isomorphism of $H$ onto $G/K$, then $G=H\oplus K.$

Thank you in advance

Arturo Magidin
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Seth
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    See also wikipedia. In your example, the surjective homomorphism $\pi\colon g\mapsto gK$ is assumed to induce an isomorphism $\pi_{\mid H}\colon H \xrightarrow{\cong} G/K$. – Dietrich Burde Oct 06 '22 at 09:41
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    Something "induces" something else when this latter is defined based on the former. – citadel Oct 06 '22 at 10:17
  • @DietrichBurde in the Wikipedia article you linked, how does one decide what the canonical map is suppose to be? – Seth Oct 06 '22 at 10:42
  • @Devo can you elaborate more about how one can tell when the something later is defined based on the former? – Seth Oct 06 '22 at 10:43
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    Seth, this depends on the context. For groups, one could answer:"you need to take the only reasonable choice". Example: The injective ring homomorphism $\iota\colon \Bbb Z\rightarrow \Bbb Q$ induces an isomorphism $\Bbb Z\cong \iota(\Bbb Z)$". By the homomorphism lemma, $G/\ker(\iota)\cong \iota(G)$, so we only have one choice, because $\ker(\iota)$ is trivial, since $\iota$ is injective. – Dietrich Burde Oct 06 '22 at 10:44
  • For example, the restriction of a map to a subset of the domain is a distinct map from the "parent" one, but clearly "induced" by it. – citadel Oct 06 '22 at 10:47
  • @Devo so an inclusion map is induced by restricting the domain of a parent function? – Seth Oct 06 '22 at 10:50
  • @DietrichBurde what you are saying is that a map can be insured when including extra machinery in the form of the homomorphism lemma in conjunction with the original homomorphism. So an isomorphism is induced by a surjective homomorphism if the domain of the original map by its kernel is isomorphic to its homomorphic image. – Seth Oct 06 '22 at 10:54
  • Take a map $f\colon X\to Y$ and a subset $S\subset X$. Then $g\colon S\to Y$ defined by $g(s):=f(s)$ is "induced by $f$". More informatively, $g=f_{|S}$. – citadel Oct 06 '22 at 11:00
  • @Devo isn't that restriction of a function? Also, when using the word "induced" or the phrase "induced homomorphism", do I have to also state by what additional theorem/lemma along with the original homomorphism that the new homomorphism is created from? – Seth Oct 06 '22 at 11:07
  • Seth, the homomorphism lemma is no extra machinery here, but just the context. If you are saying, a natural number "induces" a prime factorization, then the definition of primes is no extra machinery, but just the necessary context. – Dietrich Burde Oct 06 '22 at 11:08
  • @DietrichBurde I am getting the impression that the new homomorphism is created from the original one when I see the word "induces". Am I right that the new homomorphism can be created because of some mathematical reason? – Seth Oct 06 '22 at 11:11
  • Yes, and restriction is an example of "induction". Just an example. Try to formally define the isomorphism in your post, and I expect that its definition will involve $\operatorname{nat}_K$. So, $\operatorname{nat}_K$ "induces" that isomorphism. – citadel Oct 06 '22 at 11:11
  • @Devo the isomorphism needed involves the restriction $G$ to $H$. – Seth Oct 06 '22 at 11:13
  • @ArturoMagidin I know in category theory, it is explain in a more technical manner. I am not that advanced in category theory and I am trying to ask for explanations that can be given in non categorical language if that is possible. – Seth Oct 06 '22 at 15:31
  • @Seth why are you asking me? All I did was edit the post to correct your misuse of "it's" and of $\otimes$. I'm not in the conversations you are having. – Arturo Magidin Oct 06 '22 at 15:39
  • @ArturoMagidin ah kk. Sorry. I thought you were the one who closed the post. My bad. – Seth Oct 06 '22 at 15:49

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