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28 October, 15:18

Let G be a group. Suppose that the class equation for G is 1 + 4 + 5 + 5 + 5. (a) Does G have a subgroup of order 5? If so, then is it normal in G? Prove your answers. (b) Does G have a subgroup of order 4? If so, then is it normal in G? Prove your answers.

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  1. 28 October, 16:01
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    Answer: (a) G have a subgroup of order 5 but it is not normal in G.

    (2) G have a subgroup of order 4 but it is normal in G.

    Step-by-step explanation:

    Since we have given that

    the Class equation of G is 1+4+5+5+5.

    So, the order of G i. e. |G|=1+4+5+5+5=20

    Since we can see that there are three conjugacy classes of order 5.

    (a) So, G have a subgroup of order 5.

    Since number of subgroups of order 5 = 3

    But 3 does not divide 20.

    so, it is not normal in G.

    (b) G have a subgroup of order 4.

    Since there are only one subgroup of order 4.

    and 1 divides 20

    so, it is normal in G.
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