Isomorphisms

Find an isomorphism from the group of integers under addition to the group of even integers under addition.

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

Let $\mathbb {R}^+$ be the group of positive real numbers under multiplication. Show that the mapping $\phi (x) = \sqrt{x}$ is an automorphism of $\mathbb {R}^+$.

Show that $U(8)$ is not isomorphic to $U(10)$. Show that $U(8)$ is isomorphic to $U(12)$.

Let $G = \{ 0, \pm 2, \pm 4, \pm 6, \ldots \}$ and $H = \{ 0, \pm 3, \pm 6, \pm 9, \ldots \}$. Prove that $G$ and $H$ are isomorphic groups under addition by defining a mapping that has the required properties. Does your isomorphism preserve multiplication? Generalize to the case when $G = \left\langle m\right\rangle$ and $H = \left\langle n\right\rangle$, where $m$ and $n$ are integers.

Prove that isomorphism is an equivalence relation. That is, for any groups $G$, $H$, and $K$

Give three reasons why $S_4$ is not isomorphic to $D_{12}$.

Show that the mapping $a \to \log_{10} a$ is an isomorphism from $\mathbb {R}^+$ under multiplication to $\mathbb {R}$ under addition.

In the notation of Theorem 6.5, prove that $T_e$ is the identity and that $(T_g)^{-1} = T_{g^{-1}}$.

(Recall the notation from Cayley's Theorem: for $g \in G$, $T_g$ is the permutation of $G$ given by $T_g(x) = gx$ for all $x \in G$.)

Given that $\phi$ is a isomorphism from a group $G$ under addition to a group $\overline{G}$ under addition, convert property 2 of Theorem 6.2 to additive notation.

(Property 2 of Theorem 6.2: $G$ is Abelian if and only if $\overline{G}$ is Abelian.)

Let $G$ be a group under multiplication, $\overline{G}$ be a group under addition and $\phi$ be an isomorphism from $G$ to $\overline{G}$. If $\phi (a) = \overline{a}$ and $\phi (b) = \overline{b}$, find an expression for $\phi (a^3 b^{-2})$ in terms of $\overline{a}$ and $\overline{b}$.

Let $G$ be a group. Prove that the mapping $\alpha (g) = g^{-1}$ for all $g$ in $G$ is an automorphism if and only if $G$ is Abelian.

If $g$ and $h$ are elements from a group, prove that $\phi_g \phi_h = \phi_{gh}$.

Prove or disprove that $U(6) \approx \mathbb {Z}_6$.

Prove the assertion in Example 12 that the inner automorphisms $\phi_{R_0}, \phi_{R_{90}}, \phi_H$ and $\phi_D$ of $D_4$ are distinct.

Find two groups $G$ and $H$ such that $G \not\approx H$, but $\mathrm{Aut}(G) \approx \mathrm{Aut}(H)$.

If $G$ is a group, prove that $\mathrm{Aut}(G)$ and $\mathrm{Inn}(G)$ are groups. (This exercise is referred to in this chapter.)

If a group $G$ is isomorphic to $H$, prove that $\mathrm{Aut}(G)$ is isomorphic to $\mathrm{Aut}(H)$.

If $\phi_a$ is the inner automorphism of $G$ induced by $a$ (that is, $\phi_a(x) = axa^{-1}$ for all $x$ in $G$), prove that $(\phi_a)^n = \phi_{a^n}$ for all integers $n$.

For any group $G$ and inner automorphism $\phi_a$ of $G$ prove $|\phi_a| = |a|$.

Suppose $\phi$ belongs to $\mathrm{Aut}(\mathbb {Z}_n)$ and $a$ is relatively prime to $n$. If $\phi (a) = b$, determine a formula for $\phi (x)$.

Let $H$ be the subgroup of all rotations in $D_n$ and let $\phi$ be an automorphism of $D_n$. Prove that $\phi (H) = H$. (In words, an automorphism of $D_n$ carries rotations to rotations.)

Let $H = \left\{ \beta \in S_5 \mid \beta (1) = 1\right\}$ and $K = \left\{ \beta \in S_5 \mid \beta (2) = 2\right\}$. Prove that $H$ is isomorphic to $K$. Is the same true if $S_5$ is replaced by $S_n$, where $n \geq 3$?

Show that $\mathbb {Z}$ has infinitely many subgroups isomorphic to $\mathbb {Z}$.

Let $n$ be an even integer greater than 2 and let $\phi$ be an automorphism of $D_n$. Determine $\phi (R_{180})$.

Suppose that $G$ is an Abelian group and $\phi$ is an automorphism of $G$. Prove that $H = \left\{ x \in G \mid \phi (x) = x^4\right\}$ is a subgroup of $G$. If $\phi (x) = x^{-1}$, what can you say about the order of elements of $H$? Does your proof work when 4 is replaced by any integer $n$?

Give an example of a cyclic group of smallest order that contains both a subgroup isomorphic to $\mathbb {Z}_{12}$ and a subgroup isomorphic to $\mathbb {Z}_{20}$. Give an example of a dihedral group that has a subgroup isomorphic to $\mathbb {Z}_{12}$ and a subgroup isomorphic to $\mathbb {Z}_{20}$. No need to prove anything but explain your reasoning.

Suppose that $\phi : \mathbb {Z}_{20} \to \mathbb {Z}_{20}$ is an automorphism and $\phi (5) = 5$. What are the possibilities for $\phi (x)$?

Identify a group $G$ that has subgroups isomorphic to $\mathbb {Z}_n$ for all positive integers $n$.

Prove that every automorphism $\alpha$ of $\mathbb {R}^*$, the group of real numbers under multiplication, satisfies $\alpha (-1) = -1$.

Let $r \in U(n)$. Prove that the mapping $\alpha : \mathbb {Z}_n \to \mathbb {Z}_n$ defined by $\alpha (s) = sr \bmod n$ for all $s$ in $\mathbb {Z}_n$ is an automorphism of $\mathbb {Z}_n$. (This exercise is referred to in this chapter.)

The group $\left\{ \begin{bmatrix} 1 & a \\ 0 & 1 \end{bmatrix} \mid a \in \mathbb {Z}\right\}$ is isomorphic to what familiar group? What if $\mathbb {Z}$ is replaced by $\mathbb {R}$?

If $\phi$ and $\gamma$ are isomorphisms from the cyclic group $\left\langle a\right\rangle$ to some group and $\phi (a) = \gamma (a)$, prove that $\phi = \gamma$.

Suppose that $\phi : \mathbb {Z}_{50} \to \mathbb {Z}_{50}$ is an automorphism with $\phi (7) = 13$. Determine a formula for $\phi (x)$.

Find an isomorphism from $\mathbb {Z}_{10}$ to $U(11)$.

Let $\phi$ be an automorphism of $\mathbb {Q}^*$. Explain why $\phi$ is completely determined by the values of $\phi (p)$ for every prime $p$.

Referring to Theorem 6.5, prove that $T_g$ is indeed a permutation on the set $G$.

(Recall the notation from Cayley's Theorem: for $g \in G$, $T_g$ is defined by $T_g(x) = gx$ for all $x \in G$.)

Prove or disprove that $U(20)$ and $U(24)$ are isomorphic.

Show that the mapping $\phi (a + bi) = a - bi$ is an automorphism of the group of complex numbers under addition. Show that $\phi$ preserves complex multiplication as well---that is, $\phi (xy) = \phi (x)\phi (y)$ for all $x$ and $y$ in $\mathbb {C}$. (This exercise is referred to in Chapter 15.)

Let

and

Show that $G$ and $H$ are isomorphic under addition. Prove that $G$ and $H$ are closed under multiplication. Does your isomorphism preserve multiplication as well as addition? ($G$ and $H$ are examples of rings---a topic we will take up later.)

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

Explain why $S_8$ contains subgroups isomorphic to $\mathbb {Z}_{15}$, $U(16)$, and $D_8$.

Let $\mathbb {C}$ be the complex numbers and

Prove that $\mathbb {C}$ and $M$ are isomorphic under addition and that $\mathbb {C}^*$ and $M^*$, the nonzero elements of $M$, are isomorphic under multiplication.

Let $\mathbb {R}^n = \left\{ (a_1, a_2, \ldots , a_n) \mid a_i \in \mathbb {R}\right\}$. Show that the mapping $\phi : (a_1, a_2, \ldots , a_n) \to (-a_1, -a_2, \ldots , -a_n)$ is an automorphism of the group $\mathbb {R}^n$ under componentwise addition. This automorphism is called inversion. Describe the action of $\phi$ geometrically.

Consider the following statement: The order of a subgroup divides the order of the group. Suppose you could prove this for finite permutation groups. Would the statement then be true for all finite groups? Explain.

Suppose that $G$ is a finite Abelian group and $G$ has no element of order 2. Show that the mapping $g \to g^2$ is an automorphism of $G$. Show, by example, that there is an infinite Abelian group for which the mapping $g \to g^2$ is one-to-one and operation-preserving but not an automorphism.

Let $G$ be a group and let $g \in G$. If $z \in Z(G)$, show that the inner automorphism induced by $g$ is the same as the inner automorphism induced by $zg$ (that is, that the mappings $\phi_g$ and $\phi_{zg}$ are equal).

Prove that $\mathbb {R}$ under addition is not isomorphic to $\mathbb {R}^*$ under multiplication.

Suppose that $g$ and $h$ induce the same inner automorphism of a group $G$. Prove that $h^{-1}g \in Z(G)$. Combine this exercise with the previous one to create a single ``if and only if'' theorem.

Give two examples of nonidentity automorphisms of $S_6$.

If $\alpha$ and $\beta$ are elements in $S_n$ ($n \geq 3$), prove that $\phi_\alpha = \phi_\beta$ implies that $\alpha = \beta$. (Here, $\phi_\alpha$ is the inner automorphism of $S_n$ induced by $\alpha$.)

Prove or disprove that the mapping $\phi$ from $\mathbb {Q}^+$, the positive rational numbers under multiplication, to itself given by $\phi (x) = x^2$ is an automorphism.

Suppose the $\phi$ and $\gamma$ are isomorphisms of some group $G$ to the same group. Prove that $H = \left\{ g \in G \mid \phi (g) = \gamma (g)\right\}$ is a subgroup of $G$.

Let $G$ be a group. Complete the following statement: $|\mathrm{Inn}(G)| = 1$ if and only if .

In $S_6$ use the subgroup $H = \{ (1), (135), (153), (35)(26), (15) (26), (13)(26)\}$ to find two other subgroups of order 6 in $S_6$.

Let $\phi$ be an automorphism of $D_8$. What are the possibilities for $\phi (R_{45})$?

Let $\phi$ be an automorphism of $\mathbb {C}^*$, the group of nonzero complex numbers under multiplication. Determine $\phi (-1)$. Determine the possibilities for $\phi (i)$.

If $G$ is a non-Abelian group, prove that $G$ has an automorphism that is not the identity.

Give three examples of groups of order 120, no two of which are isomophic. Explain why they are not isomorphic.

Let $\phi$ be an automorphism of $D_4$ such that $\phi (H) = D$. Find $\phi (V)$.

Suppose that $\phi$ is an automorphism of $D_4$ such that $\phi (R_{90}) = R_{270}$ and $\phi (V) = V$. Determine $\phi (D)$ and $\phi (H)$. Determine $\phi (D)$ and $\phi (H)$ if $\phi (R_{90}) = R_{90}$ and $\phi (V) = V$.

In $\mathrm{Aut}(\mathbb {Z}_9)$, let $\alpha_i$ denote the automorphism that sends 1 to $i$ where $\gcd (i, 9) = 1$. Write $\alpha_5$ and $\alpha_8$ as permutations of $\{ 0, 1, \ldots , 8\}$ in disjoint cycle form.

Write the permutation corresponding to $R_{90}$ in the left regular representation of $D_4$ in cycle form.

Suppose that the mapping $\phi (x) = x^2$ is an automorphism of a group $G$. What can you say about the order of elements in $G$? What can you say if $\phi (x) = x^n$ for some $n > 1$ is an automorphism of $G$?

Let $F$ be a reflection in $D_n$. Explain why any automorphism $\alpha$ of $D_n$ is completely determined by the choices for $\alpha (R_{360/n})$ and $\alpha (F)$. Explain why this means that $|\mathrm{Aut}(D_n)|$ is at most $n|U(n)|$. (With some extra effort one can show $|\mathrm{Aut}(D_n)| = n|U(n)|$.)

Prove that for odd $n$, $|\mathrm{Inn}(D_n)| = 2n$ and for even $n$, $|\mathrm{Inn}(D_n)| = n$.

Let $H = \left\{ \alpha \in A_5 \mid \alpha (3) = 3\right\}$. To what familiar group is $H$ isomorphic? Let $H = \left\{ \alpha \in A_7 \mid \alpha (1) = 1, \alpha (3) = 3, \alpha (5) = 5\right\}$. To what familiar group is $H$ isomorphic?

Show that every automorphism $\phi$ of the rational numbers $\mathbb {Q}$ under addition to itself has the form $\phi (x) = x\phi (1)$.

Prove that $\mathbb {Q}^+$, the group of positive rational numbers under multiplication, is isomorphic to a proper subgroup of itself.

Prove that $\mathbb {Q}$, the group of rational numbers under addition, is not isomorphic to a proper subgroup of itself.

Prove that every automorphism of $\mathbb {R}^*$, the group of nonzero real numbers under multiplication, maps positive numbers to positive numbers and negative numbers to negative numbers.

Prove that $\mathbb {Q}^*$, the group of nonzero rational numbers under multiplication, is not isomorphic to $\mathbb {Q}$, the group of rational numbers under addition.

Give a group theoretic proof that $\mathbb {Q}$ under addition is not isomorphic to $\mathbb {R}^+$ under multiplication.

Prove or disprove that $\mathbb {Q}^*$ under multipliction is isomorphic to $\mathbb {Q}^+$ under multipliction.

For all $n \geq 1$ prove that $S_n$ is isomorphic to a subgroup of $A_{n+2}$. Why does this not contradict Theorem 5.5? This exercise is referred to in Chapter 24.