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\title{
Exam for ``Algebraic Structures'' (6ALGS) 
\\ 
summer semester 2025/2026
}

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\section*{Questions set no. 3}

\vskip 20pt

\textbf{1.} 

Formulate and prove the second and third homomorphism theorems for groups.


\vskip 20pt 

\textbf{2.}

Prove that the eight matrices 
$$
\pm \left(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}\right) , \quad
\pm \left(\begin{matrix} i & 0 \\ 0 & -i \end{matrix}\right) , \quad
\pm \left(\begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix}\right) , \quad
\pm \left(\begin{matrix} 0 & i \\ i & 0 \end{matrix}\right)
$$
form a subgroup of $GL_2(\mathbb C)$. List as much properties of this group
as you can.

\vskip 20pt

\textbf{3.}

Explicitly construct the addition and multiplication tables for the field 
consisting of:

1) 3 elements;
\\
2) 4 elements;
\\
3) 6 elements.

Describe all instances when one of these fields is a subfield of the other.


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