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

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\section*{Questions set no. 1}

\vskip 20pt

\textbf{1.} 

\textbf{1.1}. 
Define normal subgroup of a group. Is each subgroup normal?


\textbf{1.2}. 
Define group homomorphism, and its kernel and image. Illustrate these notions by
examples.

\textbf{1.3}.
What is the relationship between the kernel of a group homomorphism, and a
normal subgroup? Formulate the corresponding theorem and prove it.


\vskip 20pt 

\textbf{2.}

Find two finite groups of the same order that are not isomorphic. Show that they
are not isomorphic. What is the smallest number $n$ such that there exist two 
non-isomorphic groups of order $n$?

\vskip 20pt

\textbf{3.}

Recall that the center $Z(R)$ of a ring $R$ is defined as the set of all 
elements $z \in R$ which commute with any other element $x \in R$: $xz = zx$. 
Prove that if $x^2 - x \in Z(R)$ holds for every $x \in R$, then $R$ is 
commutative.


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