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\begin{document}

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

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\section*{Questions set no. 2}

\vskip 20pt


\textbf{1.} 

\textbf{1.1}. 
Give definition of the order of a group, and the order of an element of a group.
Illustrate these notions by examples.

\textbf{1.2}. 
Formulate and prove the Lagrange theorem about subgroups of a group.
Give examples illustrating the theorem.

\textbf{1.3}.
Formulate and prove some consequences of the Lagrange theorem. (Hint:
order of an element, cyclic groups).

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\textbf{2.}

Does there exist a cyclic group which is isomorphic to the direct product of two
nontrivial groups?

\vskip 20pt

\textbf{3.}

Let $M_2(\mathbb Q)$ be the ring of $2 \times 2$ matrices with rational 
coefficients, 
$R = 
\big\{\left(\begin{matrix} 0 & a \\ 0 & b \end{matrix}\right) \>|\> 
a,b \in \mathbb Q \big\}$,
and $I = \{A \in R \>|\> A^2 = 0 \}$.

\begin{enumerate}[\upshape(i)]
\item Prove that $R$ is a noncommutative subring of $M_2(\mathbb Q)$.
\item Prove that $I$ is an ideal in $R$.
\item Prove that $R/I \simeq \mathbb Q$.
\end{enumerate} 


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