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    Classification of groups of order $10$

    P-TOAFL

    Area
    Algebra
    Topics
    Classification, Semidirect Products, Sylow Theory
    Status
    draft

    Classify all groups of order \(10\).

    Solution

    Let \(|G|=10=2\cdot5\).

    Sylow’s theorem gives \[ n_5\mid2, \qquad n_5\equiv1\pmod5. \] Thus \[ n_5=1. \] So the Sylow \(5\)-subgroup \[ P\cong C_5 \] is normal.

    Let \(Q\cong C_2\) be a Sylow \(2\)-subgroup. Since \(P\cap Q=1\) and \(|PQ|=10\), one has \[ G\cong C_5\rtimes C_2. \] The action is determined by a homomorphism \[ C_2\to\operatorname{Aut}(C_5)\cong C_4. \] There are exactly two possibilities for the image:

    • the trivial action, giving \[ C_5\times C_2\cong C_{10}; \]
    • the unique nontrivial element of order \(2\) in \(C_4\), namely inversion on \(C_5\), giving the dihedral group of order \(10\).

    Therefore there are exactly two groups of order \(10\) up to isomorphism: \[ \boxed{C_{10}\quad\text{and}\quad D_{10}.} \]

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