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3E: Exercises

  • Page ID
    132487
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    Exercise \(\PageIndex{1}\)
    1. Consider the symmetric group \(S_4 \).

      1. Find all the subgroup lattice of \(S_4 \), the symmetric group of order 24. 

      2. Find the centre of \(S_4 \).

      3. Find the normalizer of each subgroup of \(S_4 \). 

      4. Find the cyclic subgroup of \(S_4 \) that has an order of 4. 

      5. Find a non-cyclic subgroup of \(S_4 \) that has order 4.

    Exercise \(\PageIndex{2}\)
    1. Consider the Dihedral group \(D_4 \).

      1. Find all the subgroup lattice of \(D_4 \), the Dihedral group of order 8.

      2. Find the centre of \(D_4 \).

      3. Find the normalizer of each subgroup \(D_4 \).

      4. Prove that every subgroup of \(D_4 \) of odd order is cyclic.

    Exercise \(\PageIndex{3}\)
    1. Prove that the following groups are non-abelian.

      1. \(S_n \), for \(n\ge 3 \).

      2. \(A_n \) for \(n \ge 4 \).

      3. \(D_n \), for \(n \ge 3 \).

    Exercise \(\PageIndex{4}\)

    Show that for \(n \ge 3 \), \(Z(S_n)=\{1\} \).

    Exercise \(\PageIndex{5}\)

    a. Find a cyclic subgroup of \(A_8 \) that has order 4.  

    b. Find a non-cyclic subgroup of \(A_8 \) that has order 4.

    Exercise \(\PageIndex{6}\)

    Suppose that \( \sigma \) and \( \tau \) in \(S_5 \) satisfying  \( \tau \sigma = (1, 5, 2, 3) \) and  \( \sigma \tau = (1, 2, 4, 5). \) If  \( \sigma(1)=2, \) find  \( \sigma \) and  \( \tau. \)

    Exercise \(\PageIndex{7}\)

    Factor the permutation \((1,2,3,4,5)(6,7)(1,3,5,7)(1,6,3) \) into disjoint cycles.

    Exercise \(\PageIndex{8}\)

    Let  \( \sigma = (k_1, k_2, \cdots, k_r) \) be an \(r- \)cycle. If  \( \gamma \in S_n, \) show that  \( \gamma \sigma \gamma^{-1} \) is also a cycle; infact \( \gamma \sigma \gamma^{-1}= (\gamma(k_1), \gamma(k_2), \cdots, \gamma(k_r)) \) 

    Exercise \(\PageIndex{9}\)

    Show that every product of \(3- \) cycles is an even permutation.

    Exercise \(\PageIndex{10}\)

    Show that every even permutation is a product of \(3- \) cycles.

    Exercise \(\PageIndex{11}\)
    1. Let \(G \) be a group. \(H=\{g^2: g\in G\}. \) Prove or disprove:  \(H \) is a subgroup. 

    (Hint: \(G=A_4. \))

    Exercise \(\PageIndex{12}\)

    a. Find an element of maximum order in \(S_5\).

    b. Find an element of maximum order in \(S_7\).

    Exercise \(\PageIndex{13}\)

    a. Let \(H\) be a subgroup of \(A_4\) containing \(K=\{e,(1,2)(3,4), (1,3)(2,4), (1,4)(2,3)\}\). Shoe that if \(H\) contains a \(3-\) cycle then \(H=A_4.\)

    b. Show that \(A_4\) has no subgroup of order \(6.\)

     


    This page titled 3E: Exercises is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Pamini Thangarajah.

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