Daily Exercises (Ch 1-5)

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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 e - - - -
a - b - - e
b - c d e -
c - d - a b
d - - - - -

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 $x$2$\neq e$ and $x$6$= e$, prove that $x$4$\neq e$ and $x$5$\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 1 2 3 4 5 6 7 8
2 2 1 8 7 6 5 4 3
3 3 4 5 6 7 8 1 2
4 4 3 2 1 8 7 6 5
5 5 6 7 8 1 2 3 4
6 6 5 4 3 2 1 8 7
7 7 8 1 2 3 4 5 6
8 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 $A$4. What arithmetic relationship do these orders have with the order of $A$4?

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 $A$$n$ 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 $\gamma$.

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