Chapter 1

Give at least two examples of a nontrivial proper subgroup of the indicated group. $$ G L(2, \mathbb{Q}) $$

Show that if every element of a group \(G\) is equal to its own inverse, then \(G\) is Abelian.

Let \(G\) be a group with no nontrivial proper subgroups. (a) Show that \(G\) must be cyclic. (b) What can you say about the order of \(G\) ?

In Exercises 19 through 25 find the maximum possible order of an element in the indicated group. $$ S_{4} $$

Give at least two examples of a nontrivial proper subgroup of the indicated group. $$ \mathrm{Q}_{8} $$

In Exercises 19 through 25 find the maximum possible order of an element in the indicated group. $$ S_{5} $$

Let \(G\) be a group and \(a \in G\). Show that \(a\) and \(a^{-1}\) generate the same cyclic subgroup \(\langle a\rangle=\left\langle a^{-1}\right\rangle\) and have the same order \(|a|=\left|a^{-1}\right|\)

Construct all possible group tables for a group \(G\) of order \(5 .\)

Find a generator of the subgroup \(6 \mathbb{Z} \cap 15 \mathbb{Z}\) of \(\mathbb{Z}\).

In Exercises 19 through 25 find the maximum possible order of an element in the indicated group. $$ s_{6} $$

Let \(G=\\{a+b \sqrt{2} \mid a, b \in Q\\}\). Show that \(G\) is a subgroup of \(\mathbb{R}\) under addition.

What is the order of \(G L\left(2, \mathbb{Z}_{2}\right) ?\)

Let \(m\) and \(n\) be integers. Find a generator for the subgroup \(m \mathbb{Z} \cap n \mathbb{Z}\) of \(\mathbb{Z}\).

In Exercises 19 through 25 find the maximum possible order of an element in the indicated group. $$ s_{7} $$

Let \(G=\left\\{n+m i \mid m, n \in \mathbb{Z}, i^{2}=-1\right\\}\). Show that \(G\) is a subgroup of \(C\) under addition.

Show that if \(G\) is a finite group of even order, then \(G\) has an element \(a \neq e\) such that \(a^{2}=e\).

In Exercises 19 through 25 find the maximum possible order of an element in the indicated group. $$ A_{5} $$

Let \(G=\\{\cos (2 k \pi / 7)+i \sin (2 k \pi / 7) \mid k \in \mathbb{Z}\\}\). Show that \(G\) is a subgroup of \(\mathbb{C}^{*}\) under multiplication. What is the order of \(G\) ?

In the dihedral groups \(D_{n}\) with \(n \geq 3\), show that we have \(\rho \tau=\tau \rho^{-1}\).

Let \(a\) and \(b\) be elements of a group \(G\) with \(|a|=14\) and \(|b|=15\). Describe the subgroup \(\langle a\rangle \cap\langle b\rangle\). Explain your answer.

In Exercises 19 through 25 find the maximum possible order of an element in the indicated group. $$ A_{6} $$

Let \(G=\left\\{a+b i \mid a, b \in \mathbb{R}, a^{2}+b^{2}=1\right\\} .\) Determine whether or not \(G\) is a subgroup of \(\mathbb{C}^{*}\) under multiplication.

Prove that a finite group is Abelian if and only if its group table is a symmetric matrix, that is, a matrix \(\left\\{a_{i j}\right\\}\) where \(a_{i j}=a_{j i}\) for all \(i\) and \(j\).

Let \(G=\langle a\rangle\) be a cyclic group of order \(20,\) and \(H\) and \(K\) two distinct nontrivial proper subgroups of \(G\) such that \(H \leq K,\) and \(a^{4} \notin K .\) Describe \(H\) and \(K\).

In Exercises 19 through 25 find the maximum possible order of an element in the indicated group. $$ A_{7} $$

For \(\theta \in \mathbb{R}\), let \(A(\theta) \in S L(2, \mathbb{R})\) be the matrix representing a rotation of \(\theta\) radians: $$ A(\theta)=\left[\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right] $$ (a) Show that \(H=\\{A(\theta) \mid \theta \in \mathbb{R}\\}\) is a subgroup of the special linear group \(S L(2, \mathbb{R})\) (b) Find the inverse of \(A(2 \pi / 3)\). (c) Find the order of \(A(2 \pi / 3)\).

Let \(G\) be a group, \(a \in G,\) and \(m, n\) relatively prime integers. Show that if \(a^{m}=e\), then there exists an element \(b \in G\) such that \(a=b^{n}\).

Let \(G=\langle a\rangle\) be a cyclic group of order \(n,\) and \(d\) a divisor of \(n\). Show that the number of elements in \(G\) of order \(d\) is \(\phi(d),\) where \(\phi\) is the Euler \(\phi\) -function.

Show that for any subgroup \(H\) of \(S_{n}\) either every element of \(H\) is an even permutation, or else exactly half of the elements of \(H\) are even permutations.

In the special linear group \(S L\left(2, \mathbb{Z}_{10}\right),\) let \(A=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right]\) (a) Calculate \(A^{3}\) and \(A^{11}\). (b) Find the order of \(A\).

Let \(G\) be a finite Abelian group such that for all \(a \in G, a \neq e\), we have \(a^{2} \neq e\). If \(a_{1}, a_{2}, \ldots, a_{n}\) are all the elements of \(G\) with no repetitions, evaluate the product \(a_{1} a_{2} \ldots a_{n}\)

In the special linear group \(S L(3, \mathbb{R}),\) for any \(a, b, c \in \mathbb{R},\) let $$ D(a, b, c)=\left[\begin{array}{lll} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{array}\right] . $$ Show that \(H=\\{D(a, b, c) \mid a, b, c \in \mathbb{R}\\}\) is a subgroup of \(S L(3, \mathbb{R}).\)

Let \(G\) be a nonempty finite set closed under an associative operation such that both the left and the right cancellation laws hold. Show that \(G\) under this operation is a group.

I.ct \(H=\left\\{\sigma \in S_{4} \mid \sigma(2)=2\right\\}\). (a) Show that \(H\) is a subgroup of \(S_{4}\) (b) What is the order of \(H ?\) (c) Find all the even permutations in \(H\).

Show that in an Abelian group \(G\) the set consisting of all elements of \(G\) of finite order is a subgroup of \(G\).

Show that the nonzero elements of \(\mathbb{Z}_{p},\) where \(p\) is a prime, form a group under multiplication \(\bmod p\).

Let \(n \geq 3, i \leq n,\) and let \(H=\left\\{\sigma \in S_{n} \mid \sigma(i)=i\right\\}\). (a) Show that \(H\) is a subgroup of \(S_{n}\). (b) What is the order of \(H ?\) (c) Find all the even permutations in \(H\).

Show that if \(H\) and \(K\) are subgroups of \(G,\) then \(H \cap K\) is also a subgroup of \(G\).

Prove that if \(p\) is prime, then \((p-1) ! \underline{=}-1 \bmod p\).

Show that for any \(n \geq 3, S_{n}\) is a non-Abelian group.

Show that if \(G\) is a group and \(a, b \in G,\) then \(\left|a b a^{-1}\right|=|b|\).

Find all the elements in \(S_{4}\) of order \(2 .\)

Show that if \(G\) is a group and \(a, b \in G,\) then \(|a b|=|b a|\).

Show that if \(\sigma \in S_{n}\) and \(l \sigma l=2,\) then \(\sigma\) is a product of disjoint 2 -cycles.

Let \(G\) be a group and \(a \in G\). Show that the centralizer \(C(a)\) is a subgroup of \(G\).

Show that if \(\sigma \in A_{n}\), then \(\sigma\) can be written as a product of 3 -cycles.

Show that if \(\sigma \in A_{n}\), then \(\sigma\) can be written as a product of 3 -cycles \((12 s)\) where \(s=3,4, \ldots, n\)

Let \(G\) be a group, \(a \in G\). Show that the centralizer \(C(a)=G\) if and only if \(a \in\) \(Z(G),\) the center of \(G\)

Find the center \(Z\left(S_{3}\right)\) of \(S_{3}\).

Show that if \(m \leq n,\) then the number of \(m-\) cycles \(\left(a_{1}, a_{2}, \ldots, a_{m}\right)\) in \(S_{n}\) is $$ n(n-1)(n-2) \ldots(n-m+1) / m $$