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### Homework 1

- Section 1.1
- Homework: 9, 18, 22, 25 and this problem:
- 37. Suppose \( G\) is a group with an even number of elements. Prove that the number of elements in \(G\) of order 2 is odd, and hence that \(G\) must contain an element of order 2. (Source: Joseph J. Rotman, "Advanced Modern Algebra", Prentice Hall, Upper Saddle River (2002))

- Hand In: 37

- Homework: 9, 18, 22, 25 and this problem:
- Section 1.2
- Homework: 1, 7, 9, 18 and this problem:
- 19. Show that the dihedral group \( D_{2n} = \langle r, s \vert r^n = s^2 = 1, rs = sr^{-1} \rangle \) is isomorphic to the multiplicative group generated by the two matrices \( A = \left[ \begin{array}{cc}0&1\\ 1&0 \end{array}\right] \) and \( B= \left[ \begin{array}{cc}\zeta&0\\ 0&\zeta^{-1} \end{array}\right]\), where \( \zeta = e^{\frac{2\pi i}{n}}\) is the (first) primitive \(n^{\mbox{th}}\) root of unity. (Adapted from John D. Dixon, "Problems in Group Theory", Dover, New York (1973))

- Hand In: 19

- Homework: 1, 7, 9, 18 and this problem:
- Section 1.3
- Homework: 2, 5, 9, 12 and this problem:
- 21. Give an example of \(\alpha, \beta, \gamma \in S_5\) with \(\alpha \neq 1\) such that \(\alpha\beta = \beta\alpha, \alpha\gamma = \gamma\alpha\) but \(\beta\gamma \neq \gamma\beta\). (Source: Joseph J. Rotman, "Advanced Modern Algebra", Prentice Hall, Upper Saddle River (2002))

- Hand In: 21

- Homework: 2, 5, 9, 12 and this problem:
- Section 1.4
- Homework: 2, 3, 7, 11 and this problem:
- 12. Let \(G = \mbox{GL}_2(\mathbb{Q})\), and let \( A = \left[ \begin{array}{cc}0&-1\\ 1&0 \end{array}\right] \) and \( B= \left[ \begin{array}{cc}0&1\\ -1&1 \end{array}\right]\). Show that \(A^4 = I = B^6\), but that \((AB)^n \neq I\) for all \(n > 0\), where \(I\) is the \(2 \times 2\) identity matrix. Conclude that \(AB\) can have infinite order even though both factors \(A\) and \(B\) have finite order. (Source: Joseph J. Rotman, "Advanced Modern Algebra", Prentice Hall, Upper Saddle River (2002))

- Hand In: 12

- Homework: 2, 3, 7, 11 and this problem:
- Section 1.5
- Homework: 1, 3
- Hand In: 1, extended as follows: why does your work show that D
_{8}and Q_{8}are not isomorphic?

- Section 1.6
- Homework: 4, 14, 17, 18, 23 and this problem:
- 27. Let \( G = \{ f\ :\ \mathbb{R} \to \mathbb{R}\ \vert\ f(x) = ax+b, \mbox{where}\ a \neq 0\}\). Show that \(G\) is a group under composition that is isomorphic to the subgroup of \(\mbox{GL}_2(\mathbb{R})\) consisting of all matrices of the form \(\left[ \begin{array}{cc}a&b\\ 0&1 \end{array}\right]\) (Source: Joseph J. Rotman, "Advanced Modern Algebra", Prentice Hall, Upper Saddle River (2002))

- Hand In: 27

- Homework: 4, 14, 17, 18, 23 and this problem:
- Section 3.1
- Homework: 21, 24, 32 and this problem:
- 44. Give an example to show that if \(H\) is a normal subgroup of \(G\), then \(G\) need not contain a subgroup isomorphic to \(G/H\). (Source: Joseph J. Rotman, "An Introduction to the Theory of Groups (Fourth Edition)", Springer, New York (1995))

- Hand In: 44

- Homework: 21, 24, 32 and this problem:
- Section 3.3
- Homework: 3, 8, 9 and this problem:
- 11. An \(n \times n\) matrix is called a permutation matrix if each row and each column contains exactly one \(1\) and all other entries are \(0\). If \(\pi \in S_n\), form an \(n \times n\) permutation matrix \(M(\pi)\) by defining \(M(\pi)_{ij}\) to be \(1\) if \(\pi(j) = i\) and \(0\) otherwise.
- Prove that the map \(\varphi\ :\ S_n \to \mbox{GL}_n(\mathbb{Q})\) given by \(\varphi(\pi) = M(\pi)\) is an injective homomorphism.
- Deduce that the set of \(n \times n \) permutation matrices is a group that is isomorphic with \(S_n\).
- How can one tell from \(M(\pi)\) whether the permutation \(\pi\) is even or odd?

- 11. An \(n \times n\) matrix is called a permutation matrix if each row and each column contains exactly one \(1\) and all other entries are \(0\). If \(\pi \in S_n\), form an \(n \times n\) permutation matrix \(M(\pi)\) by defining \(M(\pi)_{ij}\) to be \(1\) if \(\pi(j) = i\) and \(0\) otherwise.
- Hand In: 11

- Homework: 3, 8, 9 and this problem:
- Section 1.7
- Homework: 6, 8, 9, 10, 15
- Hand In: 15 (Warning!)

- Section 2.2
- Homework: 3, 6, 12, 14 and this problem:
- 15. Suppose that the group \(G\) acts on the set \(X\) (on the left). Let \(g \in G\). The \(\mbox{fixed point set}\) of \(g\) is the subset \(\mbox{Fix}(g) = \{x\in X\ |\ g\cdot x = x\}\) of \(X\). Suppose now that \(G\) and \(X\) are both finite. Show that the number of orbits of the action of \(G\) on \(X\) is equal to \[\frac{1}{|G|}\sum_{g\in G}|\mbox{Fix}(g)|. \] (Source: Derek J.S. Robinson, "Abstract Algebra (Second Edition)", De Gruyter, Berlin/Boston (2015))

- Hand In: 15

- Homework: 3, 6, 12, 14 and this problem:
- Section 4.1
- Homework: 4, 6, 7, 8, 9
- Hand In: -

- Section 4.2
- Homework: 1, 4, 5, 6, 8 and this problem:
- 15. Consider the dihedral group \(D_{2p}\) acting on the set \(\{1, 2, \ldots, p\}\) (considered as the consecutive vertices of a regular \(p\)-gon). Check the validity of the formula of Problem 15, Section 2.2 (above). (Source: Derek J.S. Robinson, "Abstract Algebra (Second Edition)", De Gruyter, Berlin/Boston (2015))

- Hand In: 15

- Homework: 1, 4, 5, 6, 8 and this problem:
- Section 4.3
- Homework: 2, 10, 11, 17, 18, 27 and this problem
- 37. a. Show that there are two conjugacy classes of 5-cycles in \(A_5\), each of which has 12 elements.
- Prove that the conjugacy classes in \(A_5\) have sizes 1, 12, 12, 15 and 20.
- Prove that every normal subgroup \(H\) of a group \(G\) is a union of conjugacy classes of \(G\), one of which is \(\{1\}\).
- Use (b) and (c) to give an alternate proof that \(A_5\) is simple. (Note this does not show that any other \(A_n\) is simple.)

- 37. a. Show that there are two conjugacy classes of 5-cycles in \(A_5\), each of which has 12 elements.
- Hand In: 37

- Homework: 2, 10, 11, 17, 18, 27 and this problem