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    Non-isomorphic groups can have isomorphic automorphism groups

    P-BIFDZ

    Area
    Algebra
    Topics
    Automorphisms, Classification, Counterexamples
    Status
    draft
    • Find two groups \(G\not\cong H\) where \(\Aut G\cong \Aut H\).
    Solution

    Take \[ G=C_3, \qquad H=C_4. \] Then \(G\not\cong H\) because they have different orders. On the other hand, \[ \Aut(C_n)\cong (\ZZ/n\ZZ)^\times. \] Therefore \[ \Aut(C_3)\cong (\ZZ/3\ZZ)^\times\cong C_2 \] and \[ \Aut(C_4)\cong (\ZZ/4\ZZ)^\times\cong C_2. \] Hence \[ \Aut(G)\cong\Aut(H) \] even though \(G\not\cong H\).

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