## Math 5110 Graduate Algebra I

### Homework 1

Due Friday September 14, in class (individually)
• 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
• 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
• 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
• 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
• Section 1.5
• Homework: 1, 3
• Hand In: 1, extended as follows: why does your work show that D8 and Q8 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
• 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
• 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?
(Slightly adapted from: Derek J.S. Robinson, "Abstract Algebra (Second Edition)", De Gruyter, Berlin/Boston (2015))
• Hand In: 11
• 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
• 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
• 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.
1. Prove that the conjugacy classes in $$A_5$$ have sizes 1, 12, 12, 15 and 20.
2. Prove that every normal subgroup $$H$$ of a group $$G$$ is a union of conjugacy classes of $$G$$, one of which is $$\{1\}$$.
3. 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.)
(Source: Joseph J/ Rotman, "A First Course in Abstract Algebra with Applications (Third Edition)", Prentice Hall, Upper Saddle River, NJ (2006))
• Hand In: 37