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Math Help - Non-isomorphic groups of order 42.

  1. #1
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    Non-isomorphic groups of order 42.

    Is the following argument correct, thanks

    I want to find 3 non-isomorphic groups of order 42.

    1) C_42 th cyclic group of order 42 is cyclic hence abelian.
    2) D_21 Dihedral group of order 42, non-abelain.
    3) C_3 \times D_7


    C_42 is non-isomorphic to the others, since it is
    abelian and the others are non-abelain.

    Remains to show that D_21 and C_3 \times D_7 are non-isomorphic.

    D_21 consists of 21 reflections of order 2 and 21 rotations of the form D^n.
    The possible orders for the rotations are divisors of 21, i.e, 1,3,7, or 21.
    So this group has no element of order 6.
    Now consider C_3 \times D_7 , C_3 has a generator c which has order 3, and D_7 has a reflection S of order 2, so in C_3 \times D_7 there is an element (c,S) of order lcm(3,2)=6, hence the two groups are non-isomorphic.
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  2. #2
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    Quote Originally Posted by peteryellow View Post
    Is the following argument correct, thanks

    I want to find 3 non-isomorphic groups of order 42.

    1) C_42 th cyclic group of order 42 is cyclic hence abelian.
    2) D_21 Dihedral group of order 42, non-abelain.
    3) C_3 \times D_7


    C_42 is non-isomorphic to the others, since it is
    abelian and the others are non-abelain.

    Remains to show that D_21 and C_3 \times D_7 are non-isomorphic.

    D_21 consists of 21 reflections of order 2 and 21 rotations of the form D^n.
    The possible orders for the rotations are divisors of 21, i.e, 1,3,7, or 21.
    So this group has no element of order 6.
    Now consider C_3 \times D_7 , C_3 has a generator c which has order 3, and D_7 has a reflection S of order 2, so in C_3 \times D_7 there is an element (c,S) of order lcm(3,2)=6, hence the two groups are non-isomorphic.
    It looks good to me. For these sort of problems there are many different ways to solve them.
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