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Prove that \(\mathbb Z \cong n \mathbb Z\) for \(n \neq 0\text{.}\)

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Prove that \(\mathbb Z \cong n \mathbb Z\) for \(n \neq 0\text{.}\)

Prove that \({\mathbb C}^\ast\) is isomorphic to the subgroup of \(GL_2( {\mathbb R} )\) consisting of matrices of the form

\begin{equation*} \begin{pmatrix} a & b \\ -b & a \end{pmatrix}. \end{equation*}Prove or disprove: \(U(8) \cong {\mathbb Z}_4\text{.}\)

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

\begin{equation*} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}. \end{equation*}Show that \(U(5)\) is isomorphic to \(U(10)\text{,}\) but \(U(12)\) is not.

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

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

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

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

\begin{equation*} a \ast b = a + b + ab. \end{equation*}Prove that \(G\) is a group under this operation. Show that \((G, *)\) is isomorphic to the multiplicative group of nonzero real numbers.

Show that the matrices

\begin{align*} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \quad \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \quad \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}\\ \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \quad \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \quad \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix} \end{align*}form a group. Find an isomorphism of \(G\) with a more familiar group of order \(6\text{.}\)

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

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

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

\begin{equation*} A = \begin{pmatrix} \omega & 0 \\ 0 & \omega^{-1} \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \end{equation*}generate a multiplicative group isomorphic to \(D_n\text{.}\)

Show that the set of all matrices of the form

\begin{equation*} \begin{pmatrix} \pm 1 & k \\ 0 & 1 \end{pmatrix}, \end{equation*}is a group isomorphic to \(D_n\text{,}\) where all entries in the matrix are in \({\mathbb Z}_n\text{.}\)

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

Find the order of each of the following elements.

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

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

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

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

Prove that \(D_4\) cannot be the internal direct product of two of its proper subgroups.

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 \({\mathbb Z} \times {\mathbb Z}\text{.}\)

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.

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

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

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 \(K\text{.}\)

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 \(G \cong H\text{.}\)

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

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

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

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

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

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

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

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

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

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

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 \(\phi( a + bi ) = a - bi\) is an isomorphism from \({\mathbb C}\) to \({\mathbb C}\text{.}\)

Prove that \(a + ib \mapsto a - ib\) is an automorphism of \({\mathbb C}^*\text{.}\)

Prove that \(A \mapsto B^{-1}AB\) is an automorphism of \(SL_2({\mathbb R})\) for all \(B\) in \(GL_2({\mathbb R})\text{.}\)

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

Find \(\aut( {\mathbb Z}_6)\text{.}\)

Find \(\aut( {\mathbb Z})\text{.}\)

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

Let \(G\) be a group and \(g \in G\text{.}\) Define a map \(i_g : G \rightarrow G\) by \(i_g(x) = g x g^{-1}\text{.}\) 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 \(\inn(G)\text{.}\)

Prove that \(\inn(G)\) is a subgroup of \(\aut(G)\text{.}\)

What are the inner automorphisms of the quaternion group \(Q_8\text{?}\) Is \(\inn(G) = \aut(G)\) in this case?

Let \(G\) be a group and \(g \in G\text{.}\) Define maps \(\lambda_g :G \rightarrow G\) and \(\rho_g :G \rightarrow G\) by \(\lambda_g(x) = gx\) and \(\rho_g(x) = xg^{-1}\text{.}\) 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 \(G\text{.}\)

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

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 \(n\text{.}\)

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

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 \(2p\text{.}\)

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

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

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

Suppose that \(G\) does not contain an element of order \(2p\) and \(P = \langle z \rangle\) is a subgroup of order \(p\) generated by \(z\text{.}\) If \(y\) is an element of order \(2\text{,}\) then \(yz = z^ky\) for some \(2 \leq k \lt p\text{.}\)

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

Suppose that \(G\) does not contain an element of order \(2p\) and \(P = \langle z \rangle\) 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 \(\{z^iy^j\mid 0\leq i \lt p, 0\leq j \lt 2\}\text{.}\)

Suppose that \(G\) does not contain an element of order \(2p\) and \(P = \langle z \rangle\) is a subgroup of order \(p\) generated by \(z\) and \(y\) is an element of order \(2\text{.}\) Prove that the product \((z^iy^j)(z^ry^s)\) 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.

This exercise is about putting Cayley's Theorem into practice. First, read and study the theorem. Realize that this result by itself is primarily of theoretical interest, but with some more theory we could get into some subtler aspects of this (a subject known as “representation theory”).

You should create these representations mostly with pencil-and-paper work, using Sage as a fancy calculator and assistant. You do not need to include all these computations in your worksheet. Build the requested group representations and then include enough verifications in Sage to prove that that your representation correctly represents the group.

Begin by building a permutation representation of the quaternions, \(Q\text{.}\) There are eight elements in \(Q\) (\(\pm 1, \pm I, \pm J, \pm K\)), so you will be constructing a subgroup of \(S_8\text{.}\) For each \(g\in Q\) form the function \(T_g\text{,}\) defined as \(T_g(x)=xg\text{.}\) Notice that this definition is the “reverse” of that given in the text. This is because Sage composes permutations left-to-right, while your text composes right-to-left. To create the permutations \(T_g\text{,}\) the two-line version of writing permutations could be very useful as an intermediate step. You will probably want to “code” each element of \(Q\) with an integer in \(\{1,2,\dots,8\}\text{.}\)

One such representation is included in Sage as `QuaternionGroup()` — your answer should look very similar, but perhaps not identical. Do not submit your answer for a representation of the quaternions, but I strongly suggest working this particular group representation until you are sure you have it right — the problems below might be very difficult otherwise. You can use Sage's `.is_isomorphic()` method to check if your representations are correct. However, do not use this as a substitute for the part of each question that asks you to investigate properties of your representation towards this end.

Build the permutation representation of \({\mathbb Z}_2\times{\mathbb Z}_4\) described in Cayley's Theorem. (Remember that this group is additive, while the theorem uses multiplicative notation.) Include the representation of

*each*of the \(8\) elements in your submitted work. Then construct the permutation group as a subgroup of a full symmetric group that is generated by exactly two of the eight elements you have already constructed. Hint: which two elements of \({\mathbb Z}_2\times{\mathbb Z}_4\) might you use to generate all of \({\mathbb Z}_2\times{\mathbb Z}_4\text{?}\) Use commands in Sage to investigate various properties of your permutation group, other than just`.list()`, to provide evidence that your subgroup is correct — include these in your submitted worksheet.Build a permutation representation of \(U(24)\text{,}\) the group of units mod 24. Again, list a representation of

*each*element in your submitted work. Then construct the group as a subgroup of a full symmetric group created with three generators. To determine these three generators, you will likely need to understand \(U(24)\) as an internal direct product. Use commands in Sage to investigate various properties of your group, other than just`.list()`, to provide evidence that your subgroup is correct — include these in your submitted worksheet.

Consider the symmetries of a 10-gon, \(D_{10}\) in your text, `DihedralGroup(10)` in Sage. Presume that the vertices of the 10-gon have been labeled \(1\) through \(10\) in order. Identify the permutation that is a \(180\) degree rotation and use it to generate a subgroup \(R\) of order \(2\text{.}\) Then identify the permutation that is a \(72\) degree rotation, and any one of the ten permutations that are a reflection of the \(10\)-gon about a line. Use these latter two permutations to generate a subgroup \(S\) of order \(10\text{.}\) Use Sage to verify that the full dihedral group is the internal direct product of the subgroups \(R\) and \(S\) by checking the conditions in the definition of an internal direct product.

We have a theorem which says that if a group is an internal direct product, then it is isomorphic to some external direct product. Understand that this does not mean that you can use the converse in this problem. In other words, establishing an isomorphism of \(G\) with an external direct product *does not prove* that \(G\) is an internal direct product.