3E: Exercises
- Page ID
- 132487
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Compute the following permutations:
- \((1435)(231)\)
- \(143)(231)^2\)
- \((1345)^{-1}(234)(123)\)
- \((1234)^4\)
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Consider the symmetric group \(S_4 \).
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Find the subgroup lattice of \(S_4 \), the symmetric group of order 24.
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Find the center of \(S_4 \).
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Find the normalizer of each subgroup of \(S_4 \).
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Find a cyclic subgroup of \(S_4\) of order 4.
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Find a non-cyclic subgroup of \(S_4 \) with order 4.
-
1. Determine which of the following permutations belong to \(A_6\):
\(
(1234),\qquad
(123)(45),\qquad
(1234)(56),\qquad
(12)(34).
\)
2. Find the order of
\(
(123)(4567).
\)
3. Express the permutation
\(
(13524)
\)
as a product of transpositions and determine its parity.
4. Prove that \(A_n\) is a subgroup of \(S_n\).
-
Consider the Dihedral group \(D_4 \).
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Find the subgroup lattice of \(D_4 \), the Dihedral group of order 8.
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Find the center of \(D_4 \).
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Find the normalizer of each subgroup \(D_4 \).
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Prove that every subgroup of \(D_4 \) of odd order is cyclic.
-
-
Prove that the following groups are non-abelian.
-
\(S_n \), for \(n\ge 3 \).
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\(A_n \) for \(n \ge 4 \).
-
\(D_n \), for \(n \ge 3 \).
-
Show that for \(n \ge 3 \), \(Z(S_n)=\{1\} \).
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.
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 \( \tau. \)
1) Factor the permutation \((1,2,3,4,5)(6,7)(1,3,5,7)(1,6,3) \) into disjoint cycles.
2) Express the permutation \((1,2,3,4,5)(6,7)(1,3,5,7)(1,6,3) \) as a product of transpositions, and decide if it is even or odd.
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 \( \gamma \sigma \gamma^{-1}= (\gamma(k_1), \gamma(k_2), \cdots, \gamma(k_r)) \)
Show that every product of \(3- \) cycles is an even permutation.
Show that every even permutation is a product of \(3- \) cycles.
Let \(G \) be a group. \(H=\{g^2: g\in G\}. \) Prove or disprove: \(H \) is a subgroup.
(Hint: \(G=A_4. \))
a. Find an element of maximum order in \(S_5\).
b. Find an element of maximum order in \(S_7\).
a. Let \(H\) be a subgroup of \(A_4\) containing \(K=\{e,(1,2)(3,4), (1,3)(2,4), (1,4)(2,3)\}\). Show that if \(H\) contains a \(3-\) cycle then \(H=A_4.\)
b. Show that \(A_4\) has no subgroup of order \(6.\)
Consider the symmetric group \(S_3\), compute
\(
\langle(12),(23)\rangle.
\)
Explain why this example shows that the generated subgroup of a nonabelian group cannot always be represented by one fixed ordering of powers of the generators.