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# LS10-22 - S UMMARY OF 10/22 L ECTURE We began by clarifying...

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S UMMARY OF 10/22 L ECTURE We began by clarifying what it means for two groups G and H to be isomorphic. It does not mean that every homomorphism from G to H is bijective; only that there exists a bijective homomorphism. For example, G G , but the trivial homomorphism mapping every element g 7→ 1 is not bijective. We covered several examples: (1) The cyclic group C 3 is not isomorphic to the cyclic group C 4 – they have different numbers of elements, so there are no bijective maps between them, homomorphisms or otherwise. (2) C 4 is not isomorphic to the Klein V group. This time, they have the same number of elements, so there are many (24) bijections from one to the other. But, none of these are homomorphisms. How do you know? Suppose φ : C 4 V were an isomorphism. Since C 4 is cyclic, it is generated by some element x of order 4. But φ ( x ) is some element of V , which means φ ( x ) 2 = 1 . What’s wrong with this? φ is a homomorphism, so φ ( x ) 2 = φ ( x 2 ) . This implies that φ ( x 2 ) = 1 = φ (1) . But x 2 6 = 1 in C 4

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