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4.6: Classification of Groups

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    132677
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     Classification of finite groups

    The following table gives the number of distinct groups of finite order. Here \(C_n \) stands for a cyclic group of order \(n \). 

    Group order

    Abelian 

    Non-abelian

    Comments

    1

    {e}

       

    2

    \(C_2=D_1 \)

      Prime p

    3

    \(C_3 \)

      Prime p

    4

    \(C_4 \) or 

    \(C_2 \times C_2\cong D_2 \cong K_4\) 

      Order of the form \(p^2\)

    5

    \(C_5 \)

       

    6

      \(C_6 \) 

    \(S_3\cong D_3 \)

    Order of the form \(pq\), p, q are prime with \(q \equiv 1 mod p\)

    7

    \(C_7 \) 

       

    8

    \(C_8 \) or 

      \(C_2 \times C_4 \) or

    \(C_2 \times C_2 \times C_2 \)

    \(D_4 \)  or

    \(Q_8 \)

    \(p^3\)

    9

    \(C_9 \) or 

    \(C_3 \times C_3 \) 

      \(p^2\)

    10

    \(C_{10} \)

    \(D_5 \)

     

    11

    \(C_{11} \)    

    12

    \(C_{12} \) \(D_6 \) or \(A_4\)  

    13

    \(C_{13} \)    

    14

    \(C_{14} \) \(D_7 \)  

    15

    \(C_{15} \)    

    16

    \(C_{16}\)

    or 

       

    17

    \(C_{17}\)    

    This page titled 4.6: Classification of Groups is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Pamini Thangarajah.

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