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

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

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: \(U(8) \cong {\mathbb Z}_4\).

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 \(U(10)\), but \(U(12)\) is not.

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

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

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

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.

Find five non-isomorphic groups of order 8.

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

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 \(D_n\).

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 \(D_n\), where all entries in the matrix are in \({\mathbb Z}_n\).

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

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

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

Prove or disprove: Every abelian group of order divisible by 3 contains a subgroup of order 3.

Prove or disprove: Every nonabelian group of order divisible by 6 contains a subgroup of order 6.

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

Prove or disprove the following assertion. Let \(G\), \(H\), and \(K\) be groups. If \(G \times K \cong H \times K\), then \(G \cong H\).

Prove or disprove: There is a noncyclic abelian group of order 51.

Prove or disprove: There is a noncyclic abelian group of order 52.

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

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

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

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

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

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 \(U(5) \cong {\mathbb Z}_4\). Can you generalize this result for \(U(p)\), 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 \({\mathbb C}\).

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

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

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

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

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

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

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

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

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

Let \(G\) be a group and \(g \in G\). Define maps \(\lambda_g :G \rightarrow G\) and \(\rho_g :G \rightarrow G\) by \(\lambda_g(x) = gx\) and \(\rho_g(x) = xg^{-1}\). Show that \(i_g = \rho_g \circ \lambda_g\) is an automorphism of \(G\). The isomorphism \(g \mapsto \rho_g\) is called the *right regular representation* of \(G\).

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

Let \(G\) and \(H\) be isomorphic groups. If \(G\) has a subgroup of order \(n\), prove that \(H\) must also have a subgroup of order \(n\).

If \(G \cong \overline{G}\) and \(H \cong \overline{H}\), show that \(G \times H \cong \overline{G} \times \overline{H}\).

Prove that \(G \times H\) is isomorphic to \(H \times G\).

Let \(n_1, \ldots, n_k\) be positive integers. Show that \begin{equation*}\prod_{i=1}^k {\mathbb Z}_{n_i} \cong {\mathbb Z}_{n_1 \cdots n_k}\end{equation*} if and only if \(\gcd( n_i, n_j) =1\) for \(i \neq j\).

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

If \(G\) is the internal direct product of \(H_1, H_2, \ldots, H_n\), prove that \(G\) is isomorphic to \(\prod_i H_i\).

Let \(H_1\) and \(H_2\) be subgroups of \(G_1\) and \(G_2\), respectively. Prove that \(H_1 \times H_2\) is a subgroup of \(G_1 \times G_2\).

Let \(m, n \in {\mathbb Z}\). Prove that \(\langle m,n \rangle = \langle d \rangle\) if and only if \(d = \gcd(m,n)\).

Let \(m, n \in {\mathbb Z}\). Prove that \(\langle m \rangle \cap \langle n \rangle = \langle l \rangle\) if and only if \(l = \lcm(m,n)\).

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

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

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

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

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

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

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

Suppose that \(G\) does not contain an element of order \(2p\). 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. Show that we can list the elements of \(G\) as \(\{z^iy^j\mid 0\leq i \lt p, 0\leq j \lt 2\}\).

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. 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 \(m, n\). Thus, conclude that there is only one possibility for a non-abelian group of order \(2p\), it must therefore be the one we have seen already, the dihedral group.