Math 470
HW is turned in via Gradescope, due Friday evening.
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Final HW due 12/1: 3.3, 3.4, 3.5, 3.7; and the following:
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A. Let \(G = \mathbb{R}^\times\) and \(H < G\) the subgroup of
positive numbers. Show that \(G/\{\pm 1\} \cong H\).
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B. Show that \(\mathbb{Z}^2/\langle (5,8) \rangle \cong \mathbb{Z}\).
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C. Show that \(\mathbb{Z}^2/\langle (16,6) \rangle \cong \mathbb{Z} \times \mathbb{Z}_2\).
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D. Let \(G\) be a group acting on a set \(X\). Show that the orbits form a partition of \(X\).
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E. Let \(G\) be a group acting on a set \(X\). Suppose \(x, y \in X\) are in the same orbit. Show that the stabilizers \(G_x, G_y\) are conjugate subgroups. Give an example where \(G_x \ne G_y\) (but \(x, y\) in the same orbit).
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F. Let \(X\) be the set of sequences of real numbers \((a_n)\), and let \(G = \mathbb{Z}\). For \(g \in G, a = (a_n) \in X\), define \(g(a)\) to be the sequence \((b_n)\) with \(b_n = a_{n+g}\) if \(n+g \ge 0\), and \(b_n = 0\) if \(n+g < 0\). Show that this is not a group action.
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HW due 11/10: 2/78, 79, 85, 86i,ii, 90, 97; and the following:
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A. Show that \(\langle \sigma^2 \rangle \triangleleft D_8\) and compute a complete system of representatives.
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B. Let \(n \ge 2\), and let \(H = \{\alpha \in S_n : \alpha(1) = 1\}\). Show that \(H\) is not a normal subgroup of \(S_n\), and compute a complete system of representatives (for left cosets).
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C. Let \(S_\mathbb{N}\) be the set of bijections \(\mathbb{N} \to \mathbb{N}\). For each positive integer \(n\), consider \(S_n\) as a subgroup of \(S_\mathbb{N}\) in the natural way, and let \(K = \cup S_n\). Show that \(K \triangleleft S_\mathbb{N}\) but \(K \ne S_\mathbb{N}\).
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HW due 11/3: 2/65, 69, 71, 82; and the following:
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A. Show that any cyclic group of order \(n\) is isomorphic to \(\mathbb{Z}_n\). Use this to show that \(\mathbb{Z}_{11}^\times \cong \mathbb{Z}_{10}\).
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B. Find 4 normal and 4 non-normal subgroups of \(D_8\).
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C. Show that if \(n \ge 2\) is a positive integer, then \(\mathbb{Z}_n \times \mathbb{Z}_n\) and \(\mathbb{Z}_{n^2}\) are not isomorphic.
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D. Show that \(\mathbb{Z}_8^\times\) is isomorphic to the Klein 4 group \(V\).
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E. Find 3 groups of order 12 so that no pair of them are isomorphic.
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HW due 10/27: 2/57, 59, 60, 68, 72; and the following:
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A. Show that if \(\phi: G \to H\) is a surjective homomorphism and \(G\) is cyclic, then \(H\) is cyclic.
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B. Compute \(Hom(\mathbb{Z}/24\mathbb{Z}, \mathbb{Z}/18\mathbb{Z})\).
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C. Let \(n\) be a positive integer. Prove that \(\mathbb{Z} \cong n\mathbb{Z}\).
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D. Show that \(\mathbb{Z}/6\mathbb{Z} \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}\).
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E. Show that \(\mathbb{Z}\) and \(\mathbb{Q}\) are not isomorphic.
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HW due 10/20: 2/47, 48, 49, 51, 55, 58, 66; and the following:
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A. Prove or give a counterexample: if \(G\) is a group with \(\#G = n\) and \(d \mid n, d > 0\), then \(\exists H < G\) with \(\# H = d\).
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B. Show that the map \(\phi: \mathbb{Z} \to GL_2(\mathbb{R})\) given by \(\phi(n) = \left[\begin{smallmatrix} 1 & n \\ 0 & 1 \end{smallmatrix}\right]\) is a homomorphism.
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C. Let \(G = (\mathbb{Z}, \ast)\) be the group from number 7 of the exam. Show that the map \(\phi: G \to \mathbb{Z}\) given by \(\phi(x) = x+1\) is a homomorphism.
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HW due 10/13: 2/38, 43, and the following:
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A. Let \(H \subset S_n\) be the set of \(\alpha \in S_n\) such that \(\alpha(1) = 1\). Show that \(H < S_n\).
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B. Let \(P\) be the set of polynomials with real coefficients, and let \(E \subset P\) be the even polynomials. Show that \(P\) is a group under addition, and \(E < P\).
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C. Let \(G\) be a group and \(g\) an element of finite order \(n\). For \(k \in \mathbb{Z}\), show that the order of \(g^k\) is \(n/\gcd(k,n)\). State a corollary for the case \(k = -1\).
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D. Give an example to show that if \(G\) is a group and \(H \subset G\) is the set of elements of order 2, then it is not necessarily the case that \(H < G\).
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E. In \(D_{2n}\), show that the set of rotations forms a subgroup. What about the set of reflections?
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F. In \(D_{2n}\), describe the set of elements with period 3.
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No HW due 10/6. For practice: 2/38, 39, 41, 44.
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HW due 9/29: 2/19, 22, 29, 32–36; and the following:
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A. Prove Proposition 2.51 (Laws of Exponents).
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B. Find a group \(G\) with finite cardinality \(n\), but which has no elements of order \(n\). (Any \(n\) of your choice is fine.)
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C. Let \(n\) be a positive integer and \(a \in \mathbb{Z}_n\). Give a formula for the order of \(a\) in terms of \(a\) and \(n\), and prove your answer.
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HW due 9/22: 2/21, 26i,ii,iii, 27, 28, 31
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HW due 9/15: 1/52, 73, 74; and 2/8, 13, 15, 17; and the following:
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A. How many functions are there from \(\{1, 2, 3, 4\}\) to itself?
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B. Same question, but now only count injective functions.
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C. Same question, but now only count bijective functions.
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HW due 9/8: 1/4, 6, 17, 18, 47, 51, 54, 60
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