4E: Exercises

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Nov 29, 2024
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- 4.6: Classification of Groups
- Chapter 5: Rings and fileds

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Exercise $\PageIndex{1}$

Find the number of inner automorphisms of $D_4$ .

$D_4=\{1,r,r^2,r^3,s,rs,r^2s,r^3s\}$ .

Exercise $\PageIndex{2}$

Find two groups $G_1$ and $G_2$ such that $G_1 \not \equiv G_2$ , but $\text{Aut} (G_1) \equiv \text{Aut}(G_2)$ .

Exercise $\PageIndex{3}$

Prove or disprove $U(5)\equiv \mathbb{Z}_4$ .

Exercise $\PageIndex{4}$

Find the five non-isomorphic groups of order 8.

Exercise $\PageIndex{5}$

Prove or disprove: $S_4$ is isomorphic to $D_{12}.$

Exercise $\PageIndex{6}$

Check the following structures for isomorphism.

$\mathbb{Z}_4$ under addition.
$\mathbb{Z}_5^*$ under multiplication.
$\mathbb{Z}_8^*$ under multiplication.
$\{ \pm 1, \pm i\}$ under multiplication.
Rotations that map a brick to itself.
The four functions $f(x)=x, g(x)=-x, h(x)= \dfrac{1}{x}, j(x)= \dfrac{-1}{x}$ under composition.
The four matrices $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}, \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$

Under multiplication.

Exercise $\PageIndex{7}$

Let $G=\langle a \rangle$ and $|a|=6.$ Define a mapping $\phi: G \to G$ by $\phi(g)=g^2$ for all $g \in G.$ .

1. Show that $\phi$ is a homomorphism.

2. Find $Ker(\phi).$

3. Find $Im(\phi).$

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Exercise 4E.1\PageIndex{1}

Exercise 4E.2\PageIndex{2}

Exercise 4E.3\PageIndex{3}

Exercise 4E.4\PageIndex{4}

Exercise 4E.5\PageIndex{5}

Exercise 4E.6\PageIndex{6}

Exercise 4E.7\PageIndex{7}

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