Challenging +1.8 This is a group theory proof question requiring understanding of subgroup criteria (closure, inverses, identity). The first part is moderately challenging as students must systematically verify all products in the Cayley table. The second part requires finding a counterexample to closure by computing products like a∘b, which demands careful algebraic manipulation and strategic thinking. While the concepts are A-level accessible, the abstract nature of group theory and need for rigorous proof elevates this significantly above typical A-level questions.
A group \(G\) has distinct elements \(e, a, b, c, \ldots\), where \(e\) is the identity element and \(\circ\) is the binary operation. Prove that if
$$a \circ a = b, \quad b \circ b = a$$
then the set of elements \(\{e, a, b\}\) forms a subgroup of \(G\). [5]
Prove that if
$$a \circ a = b, \quad b \circ b = c, \quad c \circ c = a$$
then the set of elements \(\{e, a, b, c\}\) does not form a subgroup of \(G\). [5]
A group $G$ has distinct elements $e, a, b, c, \ldots$, where $e$ is the identity element and $\circ$ is the binary operation. Prove that if
$$a \circ a = b, \quad b \circ b = a$$
then the set of elements $\{e, a, b\}$ forms a subgroup of $G$. [5]
Prove that if
$$a \circ a = b, \quad b \circ b = c, \quad c \circ c = a$$
then the set of elements $\{e, a, b, c\}$ does not form a subgroup of $G$. [5]
\hfill \mbox{\textit{Pre-U Pre-U 9795 Q11 [10]}}