Challenging +1.8 This is a substantial group theory question requiring knowledge of Lagrange's theorem, cyclic groups, direct products, and isomorphism. While the individual concepts are standard Further Maths fare, the question demands careful systematic work across multiple parts, particularly listing all elements and their orders in two different groups and making an isomorphism determination. This goes well beyond routine A-level content and requires genuine mathematical maturity.
Let \(G\) be a group of order 10. Write down the possible orders of the elements of \(G\) and justify your answer. [2]
Let \(G_1\) be the cyclic group of order 10 and let \(g\) be a generator of \(G_1\) (that is, an element of order 10). List the ten elements of \(G_1\) in terms of \(g\) and state the order of each element. [4]
The group \(G_2\) is defined as the set of ordered pairs \((x, y)\), where \(x \in \{0, 1\}\) and \(y \in \{0, 1, 2, 3, 4\}\), together with the binary operation \(\oplus\) defined by
$$(x_1, y_1) \oplus (x_2, y_2) = (x_3, y_3),$$
where \(x_3 = x_1 + x_2\) modulo 2 and \(y_3 = y_1 + y_2\) modulo 5.
List the elements of \(G_2\) and state the order of each element. [3]
State, with justification, whether \(G_1\) and \(G_2\) are isomorphic. [1]
\begin{enumerate}[label=(\roman*)]
\item Let $G$ be a group of order 10. Write down the possible orders of the elements of $G$ and justify your answer. [2]
\item Let $G_1$ be the cyclic group of order 10 and let $g$ be a generator of $G_1$ (that is, an element of order 10). List the ten elements of $G_1$ in terms of $g$ and state the order of each element. [4]
\item The group $G_2$ is defined as the set of ordered pairs $(x, y)$, where $x \in \{0, 1\}$ and $y \in \{0, 1, 2, 3, 4\}$, together with the binary operation $\oplus$ defined by
$$(x_1, y_1) \oplus (x_2, y_2) = (x_3, y_3),$$
where $x_3 = x_1 + x_2$ modulo 2 and $y_3 = y_1 + y_2$ modulo 5.
\begin{enumerate}[label=(\alph*)]
\item List the elements of $G_2$ and state the order of each element. [3]
\item State, with justification, whether $G_1$ and $G_2$ are isomorphic. [1]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2018 Q10 [10]}}