Daily Exercises (Ch 1-5)

## Exercises

0.00: Here is a template. For example, I can represent the real numbers as $\bf R$. In general, include the original problem here.

0.8: Suppose $a$ and $b$ are integers that divide the integer $c$. If $a$ and $b$ are relatively prime, show that $ab$ divides $c$. Show, by example, that if $a$ and $b$ are not relatively prime, then $ab$ need not divide $c$.

0.22: For every positive integer $n$, prove that $1+2+ \cdots +n=\frac{n(n+1)}{2}.$

1.6: In $D_n$, explain geometrically why a reflection followed by a reflection must be a rotation.

1.20: Bottle caps that are pried off typically have 22 ridges around the rim. Find the symmetry group of such a cap.

2.03: Show that {1,2,3} under multiplication modulo 4 is not a group but that {1,2,3,4} under modulo 5 is a group.

2.06: Give an example of group elements $a$ and $b$ with the property that a^{-1}ba \neq b 2.22: Give an example of a group with 105 elements. Give two examples of groups with 44 elements. 2.25: Suppose the table below is a group table. Fill in the blank entries. e a b c d e - - - - - b - - e - c d e - - d - a b - - - - - 2.26: Prove that if (ab)^2 = a^2b^2 in a group G, then ab=ba 2.28: Prove that the set of all rational numbers of the form 3^m 6^n, where m and n are integers, is a group under multiplication. 2.33: Suppose that G is a group with the property that for every choice of elements in G, axb=cxd implies ab=cd. Prove that G is Abelian. ("Mixed cancellation" implies commutativity.) 3.8: Let x belong to a group. If x2\neq e and x6= e, prove that x4\neq e and x5\neq e. What can we say about the order of x? 3.9: Show that if a is an element of a group G, then |a| \leq |G|. 3.10: Show that U(14) = <3> = <5>. [Hence, U(14) is cyclic.] Is U(14) = <11>? 3.14: Suppose that H is a proper subgroup of Z under addition and H contains 18,30,40. Determine H. 3.17: For each divisor k>1 of n, let U_k(n)={{x \in U(n)|x mod k =1}}. List the elements of U_4(20), U_5(20), U_5(30), and U_10(30). Prove that U_k(n) is a subgroup of U(n). Let H={{x \in U(10)|x mod 3 =1}}. Is H a subgroup of U(10)? Daily Assignment 9/10 U(8): For the value n=8, determine -the elements of U(n) -the order of the group U(n) and the order of each of its elements -Try to find at least one subgroup Daily Assignment 9/10 U(5): For the value n=5, determine -the elements of U(n) -the order of the group U(n) and the order of each of its elements -Try to find at least one subgroup Daily Assignment 9/10 U(7): For the value n=7, determine -the elements of U(n) -the order of the group U(n) and the order of each of its elements -Try to find at least one subgroup Daily Assignment 9/10 U(9): For the value n=9, determine -the elements of U(n) -the order of the group U(n) and the order of each of its elements -Try to find at least one subgroup 3.18: If H and K are subgroups of G, show that H \bigcap K is a subgroup of G. 3.20: Let G be a group and let a \epsilon G. Prove that C(a)=C(a^{-1}). 3.23: Suppose G is the group defined by the following Cayley table. 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 2 1 8 7 6 5 4 3 3 4 5 6 7 8 1 2 4 3 2 1 8 7 6 5 5 6 7 8 1 2 3 4 6 5 4 3 2 1 8 7 7 8 1 2 3 4 5 6 8 7 6 5 4 3 2 1 a. Find the centralizer of each member of G. b. Find Z(G). c. Find the order of each element of G.How are these orders arithmetically related to the order of the group? 3.28: Must the center of a group be Abelian? 3.51: Let (6) \begin{align} G = \{ \begin{bmatrix} a & b\\ c & d \end{bmatrix} | a, b, c, d \in \mathbb{Z} \} \end{align} under addition. Let (7) \begin{align} H = \{ \begin{bmatrix} a & b\\ c & d \end{bmatrix} \in G | a + b + c + d = 0 \}. \end{align} Prove that H is a subgroup of G. What if 0 is replaced by 1? 4.8: Let a be an element of a group and let \lvert a\rvert=15. Compute the orders of the following elements of G. a. a^3,a^6,a^9,a^{12} b. a^5, a^{10} c. a^2, a^4, a^8, a^{14} 4.9: How many subgroups does Z_{20} have? List a generator for each of the subgroups. Suppose that G=<a> and <a>=20. How many subgroups does G have? List a generator for each of the subgroups. 4.14: Suppose that a cyclic group G has exactly three subgroups: G itself, {e}, and a subgroup of order 7. What is |G|? What can you say if 7 is replaced with p where p is a prime? 4.18: If a cyclic group has an element of infinite order, how many elements of finite order does it have? 4.24: For any element a in any group G, prove that <a> is a subgroup of C(a) (the centralizer of a). 4.32: Determine the subgroup lattice for Z12 4.38: Consider the set \{4, 8, 12, 16\}. Show that this set is a group under multiplication modulo 20 by constructing its Cayley Table. What is the identity element? Is the group cyclic? If so, find all of its generators. 4.62: Let a be a group element such that |a| = 48. For each part find a divisor k of 28 such that a.) <a^{21}> = <a^k> b.) <a^{14}> = <a^k> c.) <a^{18}> = <a^k> 5.5: What is the order of the product of a pair of disjoint cycles of lengths 4 & 6? 5.6: Show that A8 contains an element of order 15. 5.10: Show that a function from a finite set S to itself is one-to-one if and only if it is onto. Is this true when S is infinite? 5.12: If \alpha is even, prove that \alpha^{-1} is even. If \alpha is odd, prove that \alpha^{-1} is odd. 5.17: Let (16) \begin{align} \alpha = \begin{bmatrix} 1 & 2 & 3 & 4 & 5 & 6\\ 2 & 1 & 3 & 5 & 4 & 6 \end{bmatrix}, \end{align} and (17) \begin{align} \beta = \begin{bmatrix} 1 & 2 & 3 & 4 & 5 & 6\\ 6 & 1 & 2 & 4 & 3 & 5 \end{bmatrix}, \end{align} Compute the following: a. \alpha^{-1} b. \beta\alpha c. \alpha\beta Write \alpha and \beta in cycle notation. 5.20: Compute the order of each member of A4. What arithmetic relationship do these orders have with the order of A4? 5.22: Let α and β belong to Sn. Prove that α-1β-1αβ is an even permutation. 5.24: How many elements of order 5 are in S7? 5.37: Suppose that B is a 10-cycle. For which integers i between 2 and 10 is Bi also a 10-cycle? 5.40: Represent the symmetry group of an equilateral triangle as a group of permutations of its vertices (See Exercise 3). 5.44: Find a cycle subgroup of A_8 that has order 4 5.45: Find a noncyclic subgroup of A_8 that has order 4. 5.47: Show that every element in An for $n \geq 3$ can be expressed as a 3-cycle or a product of three cycles.

5.48: Show that for $n \geq 3$, $Z(S_n) = \{\varepsilon\}$.

5.56: Given that $\beta$ and $\gamma$ are in $S_4$ with $\beta \gamma = (1432)$ and $\gamma \beta=(1243)$ and $\beta(1)=4$, determine $\beta$ and $\gamma$.