Abstract Algebra: Theory and Applications

1.

Prove that $\mathbb{Z}\cong n\mathbb{Z}$ for $n\ne 0\text{.}$

2.

Prove that ${\mathbb{C}}^{\ast }$ is isomorphic to the subgroup of $G{L}_2(\mathbb{R})$ consisting of matrices of the form

3.

Prove or disprove: $U(8)\cong {\mathbb{Z}}_4\text{.}$

4.

Prove that $U(8)$ is isomorphic to the group of matrices

5.

Show that $U(5)$ is isomorphic to $U(10)\text{,}$ but $U(12)$ is not.

6.

Show that the $n$th roots of unity are isomorphic to ${\mathbb{Z}}_n\text{.}$

7.

Show that any cyclic group of order $n$ is isomorphic to ${\mathbb{Z}}_n\text{.}$

8.

Prove that $\mathbb{Q}$ is not isomorphic to $\mathbb{Z}\text{.}$

9.

Let $G=\mathbb{R}\setminus \{-1\}$ and define a binary operation on $G$ by

Prove that $G$ is a group under this operation. Show that $(G,\ast )$ is isomorphic to the multiplicative group of nonzero real numbers.

10.

Show that the matrices

form a group. Find an isomorphism of $G$ with a more familiar group of order $6\text{.}$

11.

Find five non-isomorphic groups of order $8\text{.}$

12.

Prove $S_4$ is not isomorphic to $D_{12}\text{.}$

13.

Let $\omega =\mathrm{cis}(2\pi /n)$ be a primitive $n$th root of unity. Prove that the matrices

generate a multiplicative group isomorphic to $D_n\text{.}$

14.

Show that the set of all matrices of the form

is a group isomorphic to $D_n\text{,}$ where all entries in the matrix are in ${\mathbb{Z}}_n\text{.}$

15.

List all of the elements of ${\mathbb{Z}}_4\times {\mathbb{Z}}_2\text{.}$

16.

Find the order of each of the following elements.

1. $(3,4)$ in ${\mathbb{Z}}_4\times {\mathbb{Z}}_6$

2. $(6,15,4)$ in ${\mathbb{Z}}_{30}\times {\mathbb{Z}}_{45}\times {\mathbb{Z}}_{24}$

3. $(5,10,15)$ in ${\mathbb{Z}}_{25}\times {\mathbb{Z}}_{25}\times {\mathbb{Z}}_{25}$

4. $(8,8,8)$ in ${\mathbb{Z}}_{10}\times {\mathbb{Z}}_{24}\times {\mathbb{Z}}_{80}$

17.

Prove that $D_4$ cannot be the internal direct product of two of its proper subgroups.

18.

Prove that the subgroup of ${\mathbb{Q}}^{\ast }$ consisting of elements of the form $2^m{3}^n$ for $m,n\in \mathbb{Z}$ is an internal direct product isomorphic to $\mathbb{Z}\times \mathbb{Z}\text{.}$

19.

Prove that $S_3\times {\mathbb{Z}}_2$ is isomorphic to $D_6\text{.}$ Can you make a conjecture about $D_{2n}\text{?}$ Prove your conjecture.

20.

Prove or disprove: Every abelian group of order divisible by $3$ contains a subgroup of order $3\text{.}$

21.

Prove or disprove: Every nonabelian group of order divisible by 6 contains a subgroup of order $6\text{.}$

22.

Let $G$ be a group of order $20\text{.}$ If $G$ has subgroups $H$ and $K$ of orders $4$ and $5$ respectively such that $hk=kh$ for all $h\in H$ and $k\in K\text{,}$ prove that $G$ is the internal direct product of $H$ and $K\text{.}$

23.

Prove or disprove the following assertion. Let $G\text{,}$ $H\text{,}$ and $K$ be groups. If $G\times K\cong H\times K\text{,}$ then $G\cong H\text{.}$

24.

Prove or disprove: There is a noncyclic abelian group of order $51\text{.}$

25.

Prove or disprove: There is a noncyclic abelian group of order $52\text{.}$

26.

Let $\varphi :G\to H$ be a group isomorphism. Show that $\varphi (x)=e_H$ if and only if $x=e_G\text{,}$ where $e_G$ and $e_H$ are the identities of $G$ and $H\text{,}$ respectively.

27.

Let $G\cong H\text{.}$ Show that if $G$ is cyclic, then so is $H\text{.}$

28.

Prove that any group $G$ of order $p\text{,}$ $p$ prime, must be isomorphic to ${\mathbb{Z}}_p\text{.}$

29.

Show that $S_n$ is isomorphic to a subgroup of $A_{n+2}\text{.}$

30.

Prove that $D_n$ is isomorphic to a subgroup of $S_n\text{.}$

31.

Let $\varphi :G_1\to G_2$ and $\psi :G_2\to G_3$ be isomorphisms. Show that ${\varphi }^{-1}$ and $\psi \circ \varphi$ are both isomorphisms. Using these results, show that the isomorphism of groups determines an equivalence relation on the class of all groups.

32.

Prove $U(5)\cong {\mathbb{Z}}_4\text{.}$ Can you generalize this result for $U(p)\text{,}$ where $p$ is prime?

33.

Write out the permutations associated with each element of $S_3$ in the proof of Cayley's Theorem.

34.

An automorphism of a group $G$ is an isomorphism with itself. Prove that complex conjugation is an automorphism of the additive group of complex numbers; that is, show that the map $\varphi (a+bi)=a-bi$ is an isomorphism from $\mathbb{C}$ to $\mathbb{C}\text{.}$

35.

Prove that $a+ib\mapsto a-ib$ is an automorphism of ${\mathbb{C}}^{\ast }\text{.}$

36.

Prove that $A\mapsto B^{-1}AB$ is an automorphism of $S{L}_2(\mathbb{R})$ for all $B$ in $G{L}_2(\mathbb{R})\text{.}$

37.

We will denote the set of all automorphisms of $G$ by $\mathrm{Aut}(G)\text{.}$ Prove that $\mathrm{Aut}(G)$ is a subgroup of $S_G\text{,}$ the group of permutations of $G\text{.}$

38.

Find $\mathrm{Aut}({\mathbb{Z}}_6)\text{.}$

39.

Find $\mathrm{Aut}(\mathbb{Z})\text{.}$

40.

Find two nonisomorphic groups $G$ and $H$ such that $\mathrm{Aut}(G)\cong \mathrm{Aut}(H)\text{.}$

41.

Let $G$ be a group and $g\in G\text{.}$ Define a map $i_g:G\to G$ by $i_g(x)=gx{g}^{-1}\text{.}$ Prove that $i_g$ defines an automorphism of $G\text{.}$ Such an automorphism is called an inner automorphism. The set of all inner automorphisms is denoted by $\mathrm{Inn}(G)\text{.}$

42.

Prove that $\mathrm{Inn}(G)$ is a subgroup of $\mathrm{Aut}(G)\text{.}$

43.

What are the inner automorphisms of the quaternion group $Q_8\text{?}$ Is $\mathrm{Inn}(G)=\mathrm{Aut}(G)$ in this case?

44.

Let $G$ be a group and $g\in G\text{.}$ Define maps ${\lambda }_g:G\to G$ and ${\rho }_g:G\to G$ by ${\lambda }_g(x)=gx$ and ${\rho }_g(x)=x{g}^{-1}\text{.}$ Show that $i_g={\rho }_g\circ {\lambda }_g$ is an automorphism of $G\text{.}$ The isomorphism $g\mapsto {\rho }_g$ is called the right regular representation of $G\text{.}$

45.

Let $G$ be the internal direct product of subgroups $H$ and $K\text{.}$ Show that the map $\varphi :G\to H\times K$ defined by $\varphi (g)=(h,k)$ for $g=hk\text{,}$ where $h\in H$ and $k\in K\text{,}$ is one-to-one and onto.

46.

Let $G$ and $H$ be isomorphic groups. If $G$ has a subgroup of order $n\text{,}$ prove that $H$ must also have a subgroup of order $n\text{.}$

47.

If $G\cong \bar{G}$ and $H\cong \bar{H}\text{,}$ show that $G\times H\cong \bar{G}\times \bar{H}\text{.}$

48.

Prove that $G\times H$ is isomorphic to $H\times G\text{.}$

49.

Let $n_1,\dots ,n_k$ be positive integers. Show that

if and only if $gcd(n_i,n_j)=1$ for $i\ne j\text{.}$

50.

Prove that $A\times B$ is abelian if and only if $A$ and $B$ are abelian.

51.

If $G$ is the internal direct product of $H_1,H_2,\dots ,H_n\text{,}$ prove that $G$ is isomorphic to $\prod _i{H}_i\text{.}$

52.

Let $H_1$ and $H_2$ be subgroups of $G_1$ and $G_2\text{,}$ respectively. Prove that $H_1\times H_2$ is a subgroup of $G_1\times G_2\text{.}$

53.

Let $m,n\in \mathbb{Z}\text{.}$ Prove that $⟨m,n⟩=⟨d⟩$ if and only if $d=gcd(m,n)\text{.}$

54.

Let $m,n\in \mathbb{Z}\text{.}$ Prove that $⟨m⟩\cap ⟨n⟩=⟨l⟩$ if and only if $l=\mathrm{lcm}(m,n)\text{.}$

55. Groups of order $2p$.

In this series of exercises we will classify all groups of order $2p\text{,}$ where $p$ is an odd prime.

1. Assume $G$ is a group of order $2p\text{,}$ where $p$ is an odd prime. If $a\in G\text{,}$ show that $a$ must have order $1\text{,}$ $2\text{,}$ $p\text{,}$ or $2p\text{.}$

2. Suppose that $G$ has an element of order $2p\text{.}$ Prove that $G$ is isomorphic to ${\mathbb{Z}}_{2p}\text{.}$ Hence, $G$ is cyclic.

3. Suppose that $G$ does not contain an element of order $2p\text{.}$ Show that $G$ must contain an element of order $p\text{.}$ Hint: Assume that $G$ does not contain an element of order $p\text{.}$

4. Suppose that $G$ does not contain an element of order $2p\text{.}$ Show that $G$ must contain an element of order $2\text{.}$

5. Let $P$ be a subgroup of $G$ with order $p$ and $y\in G$ have order $2\text{.}$ Show that $yP=Py\text{.}$

6. Suppose that $G$ does not contain an element of order $2p$ and $P=⟨z⟩$ is a subgroup of order $p$ generated by $z\text{.}$ If $y$ is an element of order $2\text{,}$ then $yz=z^{k}y$ for some $2\le k<p\text{.}$

7. Suppose that $G$ does not contain an element of order $2p\text{.}$ Prove that $G$ is not abelian.

8. Suppose that $G$ does not contain an element of order $2p$ and $P=⟨z⟩$ is a subgroup of order $p$ generated by $z$ and $y$ is an element of order $2\text{.}$ Show that we can list the elements of $G$ as $\{z^i{y}^j\mid 0\le i<p,0\le j<2\}\text{.}$

9. Suppose that $G$ does not contain an element of order $2p$ and $P=⟨z⟩$ is a subgroup of order $p$ generated by $z$ and $y$ is an element of order $2\text{.}$ Prove that the product $(z^i{y}^j)(z^r{y}^s)$ can be expressed as a uniquely as $z^m{y}^n$ for some non negative integers $m,n\text{.}$ Thus, conclude that there is only one possibility for a non-abelian group of order $2p\text{,}$ it must therefore be the one we have seen already, the dihedral group.