| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795 (Pre-U Further Mathematics) |
| Session | Specimen |
| Topic | Groups |
| Type | Subgroups and cosets |
| Difficulty | Challenging +1.8 This is an abstract algebra proof question requiring understanding of subgroup axioms (closure, identity, inverses) and systematic verification. While the topic (group theory) is advanced Further Maths content, the proofs themselves are relatively structured once you know what to check. Part (i) requires showing closure under the operation and finding inverses; part (ii) requires finding a counterexample to closure. The conceptual demand is high for A-level, but the execution is more mechanical than questions requiring deep geometric insight or novel problem-solving strategies. |
| Spec | 8.03c Group definition: recall and use, show structure is/isn't a group8.03f Subgroups: definition and tests for proper subgroups |
10 A group $G$ has distinct elements $e , a , b , c , \ldots$, where $e$ is the identity element and $\circ$ is the binary operation.\\
(i) Prove that if $a \circ a = b$ and $b \circ b = a$, then the set of elements $\{ e , a , b \}$ forms a subgroup of $G$.\\
(ii) Prove that if $a \circ a = b , b \circ b = c$ and $c \circ c = a$, then the set of elements $\{ e , a , b , c \}$ does not form a subgroup of $G$.
\hfill \mbox{\textit{Pre-U Pre-U 9795 Q10}}