StudyGuideIsomorphisms - Isomorphisms and Automorphisms...

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Isomorphisms and Automorphisms Study Guide Outline 1. Isomorphisms Let G and H be groups. An isomorphism from G to H is a function ϕ : G H satisfying the following conditions: ϕ is a bijection, and ϕ ( xy ) = ϕ ( x ) ϕ ( y ) for all x,y G . If there exists an isomorphism from G to H , then the groups G and H are said to be iso- morphic . Problems: 3 2. Isomorphism Type If G and H are isomorphic groups, then: G and H must either both be abelian or both be non-abelian, and G and H must have the same number of elements of each order. For small groups, this makes it possible to determine the isomorphism type of a given group. A group of order n that has an element of order n is cyclic. The following table shows the isomorphism types of non-cyclic groups that we are aware of: Abelian Groups (Not Cyclic) Order Groups 4 V 8 Z 4 × Z 2 , V × Z 2 9 Z 3 × Z 3 12 Z 6 × Z 2 16 Z 8 × Z 2 , Z 4 × Z 4 , Z 4 × V, V × V Non-Abelian Groups Order Groups 6 S 3 8 D 4 , Q
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StudyGuideIsomorphisms - Isomorphisms and Automorphisms...

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