Abstract Algebra: Theory and Applications

Write out Cayley tables for groups formed by the symmetries of a rectangle and for $({\mathbb{Z}}_4,+)\text{.}$ How many elements are in each group? Are the groups the same? Why or why not?

Describe the symmetries of a rhombus and prove that the set of symmetries forms a group. Give Cayley tables for both the symmetries of a rectangle and the symmetries of a rhombus. Are the symmetries of a rectangle and those of a rhombus the same?

Describe the symmetries of a square and prove that the set of symmetries is a group. Give a Cayley table for the symmetries. How many ways can the vertices of a square be permuted? Is each permutation necessarily a symmetry of the square? The symmetry group of the square is denoted by $D_4\text{.}$

Give a multiplication table for the group $U(12)\text{.}$

Let $S=\mathbb{R}\setminus \{-1\}$ and define a binary operation on $S$ by $a\ast b=a+b+ab\text{.}$ Prove that $(S,\ast )$ is an abelian group.

Give an example of two elements $A$ and $B$ in $G{L}_2(\mathbb{R})$ with $AB\ne BA\text{.}$

Prove that the product of two matrices in $S{L}_2(\mathbb{R})$ has determinant one.

Prove that $det(AB)=det(A)det(B)$ in $G{L}_2(\mathbb{R})\text{.}$ Use this result to show that the binary operation in the group $G{L}_2(\mathbb{R})$ is closed; that is, if $A$ and $B$ are in $G{L}_2(\mathbb{R})\text{,}$ then $AB\in G{L}_2(\mathbb{R})\text{.}$

Show that ${\mathbb{R}}^{\ast }=\mathbb{R}\setminus \{0\}$ is a group under the operation of multiplication.

Given the groups ${\mathbb{R}}^{\ast }$ and $\mathbb{Z}\text{,}$ let $G={\mathbb{R}}^{\ast }\times \mathbb{Z}\text{.}$ Define a binary operation $\circ$ on $G$ by $(a,m)\circ (b,n)=(ab,m+n)\text{.}$ Show that $G$ is a group under this operation.

Prove or disprove that every group containing six elements is abelian.

Give a specific example of some group $G$ and elements $g,h\in G$ where $(gh{)}^n\ne g^n{h}^n\text{.}$

Give an example of three different groups with eight elements. Why are the groups different?

Show that there are $n!$ permutations of a set containing $n$ items.

Show that addition and multiplication mod $n$ are well defined operations. That is, show that the operations do not depend on the choice of the representative from the equivalence classes mod $n\text{.}$

Show that addition and multiplication mod $n$ are associative operations.

Let $a$ and $b$ be elements in a group $G\text{.}$ Prove that $a{b}^n{a}^{-1}=(ab{a}^{-1}{)}^n$ for $n\in \mathbb{Z}\text{.}$

Let $U(n)$ be the group of units in ${\mathbb{Z}}_n\text{.}$ If $n>2\text{,}$ prove that there is an element $k\in U(n)$ such that $k^2=1$ and $k\ne 1\text{.}$

Prove that the inverse of $g_1{g}_2\cdots g_n$ is $g_n^{-1}{g}_{n-1}^{-1}\cdots g_1^{-1}\text{.}$

Prove the remainder of Proposition 3.21: if $G$ is a group and $a,b\in G\text{,}$ then the equation $xa=b$ has a unique solution in $G\text{.}$

Prove Theorem 3.23.

Prove the right and left cancellation laws for a group $G\text{;}$ that is, show that in the group $G\text{,}$ $ba=ca$ implies $b=c$ and $ab=ac$ implies $b=c$ for elements $a,b,c\in G\text{.}$

Show that if $a^2=e$ for all elements $a$ in a group $G\text{,}$ then $G$ must be abelian.

Show that if $G$ is a finite group of even order, then there is an $a\in G$ such that $a$ is not the identity and $a^2=e\text{.}$

Let $G$ be a group and suppose that $(ab{)}^2=a^2{b}^2$ for all $a$ and $b$ in $G\text{.}$ Prove that $G$ is an abelian group.

Find all the subgroups of ${\mathbb{Z}}_3\times {\mathbb{Z}}_3\text{.}$ Use this information to show that ${\mathbb{Z}}_3\times {\mathbb{Z}}_3$ is not the same group as ${\mathbb{Z}}_9\text{.}$ (See Example 3.28 for a short description of the product of groups.)

Find all the subgroups of the symmetry group of an equilateral triangle.

Compute the subgroups of the symmetry group of a square.

Let $H=\{2^k:k\in \mathbb{Z}\}\text{.}$ Show that $H$ is a subgroup of ${\mathbb{Q}}^{\ast }\text{.}$

Let $n=0,1,2,\dots$ and $n\mathbb{Z}=\{nk:k\in \mathbb{Z}\}\text{.}$ Prove that $n\mathbb{Z}$ is a subgroup of $\mathbb{Z}\text{.}$ Show that these subgroups are the only subgroups of $\mathbb{Z}\text{.}$

Let $\mathbb{T}=\{z\in {\mathbb{C}}^{\ast }:|z|=1\}\text{.}$ Prove that $\mathbb{T}$ is a subgroup of ${\mathbb{C}}^{\ast }\text{.}$

Prove or disprove: $S{L}_2(\mathbb{Z})\text{,}$ the set of $2\times 2$ matrices with integer entries and determinant one, is a subgroup of $S{L}_2(\mathbb{R})\text{.}$

List the subgroups of the quaternion group, $Q_8\text{.}$

Prove that the intersection of two subgroups of a group $G$ is also a subgroup of $G\text{.}$

Prove or disprove: If $H$ and $K$ are subgroups of a group $G\text{,}$ then $H\cup K$ is a subgroup of $G\text{.}$

Prove or disprove: If $H$ and $K$ are subgroups of a group $G\text{,}$ then $HK=\{hk:h\in H\text{ and }k\in K\}$ is a subgroup of $G\text{.}$ What if $G$ is abelian?

Let $a$ and $b$ be elements of a group $G\text{.}$ If $a^{4}b=ba$ and $a^3=e\text{,}$ prove that $ab=ba\text{.}$

Give an example of an infinite group in which every nontrivial subgroup is infinite.

If $xy=x^{-1}{y}^{-1}$ for all $x$ and $y$ in $G\text{,}$ prove that $G$ must be abelian.

Prove or disprove: Every proper subgroup of a nonabelian group is nonabelian.

Let $H$ be a subgroup of $G\text{.}$ If $g\in G\text{,}$ show that $gH{g}^{-1}=\{gh{g}^{-1}:h\in H\}$ is also a subgroup of $G\text{.}$