1
Let \(S = \bigl\{ x \in \mathbb{R} \colon x \gt 6 \bigr\}\) and define a binary operation on \(S\) by
\begin{equation*}
a \star b = (a-1)(b-5) + 1 .
\end{equation*}
Prove that \((S,\star)\) satisfies the closure, identity, and inverse properties. Furthermore, determine (with proof) whether \((S,\star)\) is a group.
3
Let \(G\) be a finite cyclic group whose order is even. Let \(g \in G\) be an element whose order is odd.
Prove that the order of \(g\) agrees with the order of its square:
\begin{equation*}
\bigl| g^2 \bigr| = |g|.
\end{equation*}
4
Let \(G\) be a finite group whose order is even. Define a relation \(\sim\) on \(G\) by
\begin{equation*}
a \sim b \text{ if } \begin{cases}a=b \amp {\rm or}\\ ab=e\end{cases}.
\end{equation*}
Prove that \(\sim\) is an equivalence relation. Then, use this equivalence relation to prove that there must exist an element of order 2 in the group \(G\text{.}\)