| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Topic | Groups |
| Type | Proper subgroups identification |
| Difficulty | Challenging +1.8 This is a group theory question requiring construction of a Cayley table for an elementary abelian group of order 8 (isomorphic to C₂×C₂×C₂) and identification of its subgroups. While the topic is advanced (abstract algebra, typically Further Maths), the actual operations are straightforward given that all generators square to identity and commute. The subgroup enumeration requires systematic thinking but follows standard patterns. This is moderately challenging due to the abstract algebra content and careful bookkeeping required, but it's a relatively standard group theory exercise at this level. |
| Spec | 8.03b Cayley tables: construct for finite sets under binary operation8.03f Subgroups: definition and tests for proper subgroups |
| Answer | Marks |
|---|---|
| (ii) | e a b c ab bc ca abc |
| Answer | Marks |
|---|---|
| Or (5 distinct) + B1 (6th) + B1 (all 7 and no repeats) | B4, 3, 2, |
Question 8:
--- 8 (i)
(ii) ---
8 (i)
(ii) | e a b c ab bc ca abc
e e a b c ab bc ca abc
a a e ab ca b abc c bc
b b ab e bc a c abc ca
c c ca bc e abc b a ab
ab ab b a abc e ca bc c
bc bc abc c b ca e ab a
ca ca c abc a bc ab e b
abc abc bc ca ab c a b e
Subgroups of order 2: {e, a}, { e, b},{ e, c},{ e, ab},{ e, bc},{ e, ca},{ e, abc}
6 distinct given
all 7, no repeats
Subgroups of order 4: {e, a, b, ab}, {e, b, c, bc}, {e, c, a, ca}
{e, a, bc, abc}, {e, b, ca, abc}, {e, c, ab, abc}
{e, ab, bc, ca}
Either (all 3 of 1st type) + B1 (all3 of 2nd type) + B1 (last one)
Or (5 distinct) + B1 (6th) + B1 (all 7 and no repeats) | B4, 3, 2,
1, 0
[4]
B1
B1
B2
[5]The group $G$, of order 8, consists of the elements $\{e, a, b, c, ab, bc, ca, abc\}$, together with a multiplicative binary operation, where $e$ is the identity and
$$a^2 = b^2 = c^2 = e, \quad ab = ba, \quad bc = cb \quad \text{and} \quad ca = ac.$$
\begin{enumerate}[label=(\roman*)]
\item Construct the group table of $G$. [You are not required to show how individual elements of the table are determined.] [4]
\item List all the proper subgroups of $G$. [5]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2015 Q8 [9]}}