301.10H Applying the First Isomorphism Theorem

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The first isomorphism theorem helps us "whittle down" a homomorphism to discover an isomorphism within. It does this by giving us a normal subgroup of G (the kernel of the homomorphism) and also telling us what the factor group by that normal subgroup is (it's isomorphic to the image of the homomorphism). Here's an example using dihedral and symmetric groups.

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Applying the first? More like "Amazing videos that quench a thirst"...of knowledge! 👍

PunmasterSTP

How do we know to map to the transpositions (12) and (34). I kinda missed the point, how to determine to pick which elements in S4.

krumpy

The factor group is dual to the image -- the 1st isomorphism theorem.
Injective is dual to surjective synthesizes bijective or isomorphism.
Domains are dual to co-domains.
Same is dual to different -- removing the differences via projection/contraction synthesizes isomorphism or duality!

hyperduality

301.10H Applying the First Isomorphism Theorem

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