4E: Exercises

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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.