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Math Help - lagrange problem set

  1. #1
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    lagrange problem set

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  2. #2
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    For #1, let N_1,N_2 be normal subgroups of orders 3\mbox{ and }5 respectively. Using the identity |N_1N_2||N_1\cap N_2| = |N_1||N_2| and the fact that N_1\cap N_2 = \{ e\} we find that N_1N_2 is a subgroup of order 15. Let N_1N_2 = H then N_1,N_2\triangleleft H with N_1\cap N_2 = \{ e\} this means H\simeq N_1\times N_2 \simeq \mathbb{Z}_3\times \mathbb{Z}_5\simeq \mathbb{Z}_15. Therefore this group is cyclic of order 15.

    Note, D_5 has ten elements but it is not cyclic.
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  3. #3
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    For #2, let G be a group of prime order. Let a\in G so that a\not = e. Argue that \left< a \right> = G using Lagrange's theorem.
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  4. #4
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    plz help

    please help me with second question last part....
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  5. #5
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    Quote Originally Posted by szpengchao View Post
    please help me with second question last part....
    The general result says that the subgroups of \mathbb{Z}_n (the cyclic subgroup of order n) are precisely k\mathbb{Z}_n = \left< [k]_n\right> where k is a positive divisor of n. And furthermore, k\mathbb{Z}_n is a subgroup of m\mathbb{Z}_n if and only if m|k.

    For example, the complete list of subgroups of \mathbb{Z}_{24} are given below.
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  6. #6
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    help

    what is the triangle in question 1, N_1 , N_2, triangle H

    is that : belongs to?
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  7. #7
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    Quote Originally Posted by szpengchao View Post
    what is the triangle in question 1, N_1 , N_2, triangle H

    is that : belongs to?
    It means normal subgroup.
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  8. #8
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    help


    what does [k]_n means?
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