Copy and complete the table.

Show that \(\{ S,{\text{ }} * \} \) is an Abelian group.

Determine the orders of all the elements of \(\{ S,{\text{ }} * \} \).

(i)     Find the two proper subgroups of \(\{ S,{\text{ }} * \} \).

(ii)     Find the coset of each of these subgroups with respect to the element 5.

Solve the equation \(2 * x * 4 * x * 4 = 2\).

     A3

Note:     Award A3 for correct table, A2 for one or two errors, A1 for three or four errors and A0 otherwise.

[3 marks]

the table contains only elements of \(S\), showing closure     R1

the identity is 1     A1

every element has an inverse since 1 appears in every row and column, or a complete list of elements and their correct inverses     A1

multiplication of numbers is associative     A1

the four axioms are satisfied therefore \(\{ S,{\text{ }} * \} \) is a group

the group is Abelian because the table is symmetric (about the leading diagonal)     A1

[5 marks]

     A3

Note:     Award A3 for all correct values, A2 for 5 correct, A1 for 4 correct and A0 otherwise.

[3 marks]

(i)     the subgroups are \(\{ 1,{\text{ }}8\} \); \(\{ 1,{\text{ }}4,{\text{ }}7\} \)     A1A1

(ii)     the cosets are \(\{ 4,{\text{ }}5\} \); \(\{ 2,{\text{ }}5,{\text{ }}8\} \)     A1A1

[4 marks]

METHOD 1

use of algebraic manipulations     M1

and at least one result from the table, used correctly     A1

\(x = 2\)    A1

\(x = 7\)    A1

METHOD 2

testing at least one value in the equation     M1

obtain \(x = 2\)     A1

obtain \(x = 7\)     A1

explicit rejection of all other values     A1

[4 marks]

The majority of candidates were able to complete the Cayley table correctly.

Generally well done. However, it is not good enough for a candidate to say something along the lines of 'the operation is closed or that inverses exist by looking at the Cayley table'. A few candidates thought they only had to prove commutativity.

Often well done. A few candidates stated extra, and therefore incorrect subgroups.

The majority found only one solution, usually the obvious \(x = 2\), but sometimes only the less obvious \(x = 7\).