AS June 2019 Q4

EdexcelCurrent spec7 marksGroups

4. The set \(\{e, p, q, r, s\}\) forms a group, \(A\), under the operation \(*\)

Given that \(e\) is the identity element and that

\[p * p = s \qquad s * s = r \qquad p * p * p = q\]

(a) show that

(i) \(p * q = r\)

(ii) \(s * p = q\)

(2)

(b) Hence complete the Cayley table below.

(2)

A student states that there is a subgroup of \(A\) of order 3

(d) Comment on the validity of this statement, giving a reason for your answer. (2)

Mark scheme (a)

Scheme	Marks	AO
\(p * q = p * p * p * p = s * s = r\) OR \(s * s = r \Rightarrow p * p * p * p = r \Rightarrow p * q = r\)	B1	2.1
\(s * p = p * p * p = q\) OR as \(p * p * p = q\) and \(p * p = s \Rightarrow s * p = q\)	B1	2.1
	(2)

B1: Correct proof to achieve the printed statement

Mark scheme (b)

Scheme

Marks

\(*\)	\(e\)	\(p\)	\(q\)	\(r\)	\(s\)
\(e\)	\(\boldsymbol{e}\)	\(\boldsymbol{p}\)	\(\boldsymbol{q}\)	\(\boldsymbol{r}\)	\(\boldsymbol{s}\)
\(p\)	\(\boldsymbol{p}\)	\(\boldsymbol{s}\)	\(\boldsymbol{r}\)	\(e\)	\(q\)
\(q\)	\(\boldsymbol{q}\)	\(r\)	\(p\)	\(s\)	\(e\)
\(r\)	\(\boldsymbol{r}\)	\(e\)	\(s\)	\(q\)	\(p\)
\(s\)	\(\boldsymbol{s}\)	\(\boldsymbol{q}\)	\(e\)	\(p\)	\(\boldsymbol{r}\)

M1
A1

1.1b
1.1b

(2)

(b) Marked B1 B1 on ePen

M1: Finds at least 13 correct entries – usually the highlighted

A1: Completely correct table

(The entries highlighted in the printed mark scheme are shown in bold.)

Scheme	Marks	AO
\(p * q * r * s = e\)	B1	1.1b
	(1)

Mark scheme (d)

Scheme	Marks	AO
The order of a subgroup is a factor of the order of the group (Lagrange’s Theorem)	M1	1.2
As 3 is not a factor of 5, the student’s statement is wrong	A1	2.3
	(2)
	(7 marks)

M1: Some indication that the order of a subgroup must be a factor of the order of the group. May say that 3 is not a factor of 5 or equivalent

A1: Fully correct unambiguous statement that refers Lagrange’s theorem and either

and comments that the student’s statement is incorrect. No contradictory statements