AS June 2023 Q1

EdexcelCurrent spec8 marksGroups

1. The operation \(*\) is defined on the set \(G = \{0, 1, 2, 3\}\) by

\[x * y \equiv x + y - 2xy \pmod{4}\]

(a) Complete the Cayley table below.

(2)

(b) Show that \(G\) is a group under the operation \(*\)
(You may assume the associative law is satisfied.) (3)

(d) State whether \(G\) is a cyclic group, giving a reason for your answer. (1)

Mark scheme (b)

Scheme	Marks	AO
Identity element is 0 and there is closure	M1	2.1
0 is the identity so is self-inverse 1, 2 and 3 are self-inverse	M1	2.5
Associative law is assumed so \(G\) forms a group	A1	1.1b
	(3)

M1: States closure and identifies 0 as the identity element.

M1: Finds inverses for each element.

A1: Scores both previous method marks and states that the associative law is satisfied therefore \(G\) is a group

Scheme	Marks	AO
0 has order 1 and 1, 2 and 3 have order 2	M1 A1	1.1b 1.1b
	(2)

Mark scheme (d)

Scheme	Marks	AO
There is no element with order 4 therefore \(G\) is not a cyclic group Every element is its own inverse therefore no group generator therefore \(G\) is not a cyclic group	B1	2.4
	(1)
	(8 marks)

B1: See scheme