Date | May 2018 | Marks available | 3 | Reference code | 18M.3srg.hl.TZ0.1 |
Level | HL only | Paper | Paper 3 Sets, relations and groups | Time zone | TZ0 |
Command term | Find | Question number | 1 | Adapted from | N/A |
Question
The binary operation multiplication modulo 10, denoted by ×10, is defined on the set T = {2 , 4 , 6 , 8} and represented in the following Cayley table.
Show that {T, ×10} is a group. (You may assume associativity.)
By making reference to the Cayley table, explain why T is Abelian.
Find the order of each element of {T, ×10}.
Hence show that {T, ×10} is cyclic and write down all its generators.
The binary operation multiplication modulo 10, denoted by ×10 , is defined on the set V = {1, 3 ,5 ,7 ,9}.
Show that {V, ×10} is not a group.
Markscheme
closure: there are no new elements in the table A1
identity: 6 is the identity element A1
inverse: every element has an inverse because there is a 6 in every row and column (2−1 = 8, 4−1 = 4, 6−1 = 6, 8−1 = 2) A1
we are given that (modulo) multiplication is associative R1
so {T, ×10} is a group AG
[4 marks]
the Cayley table is symmetric (about the main diagonal) R1
so T is Abelian AG
[1 mark]
considering powers of elements (M1)
A2
Note: Award A2 for all correct and A1 for one error.
[3 marks]
EITHER
{T, ×10} is cyclic because there is an element of order 4 R1
Note: Accept “there are elements of order 4”.
OR
{T, ×10} is cyclic because there is generator R1
Note: Accept “because there are generators”.
THEN
2 and 8 are generators A1A1
[3 marks]
EITHER
considering singular elements (M1)
5 has no inverse (5 ×10 a = 1, a∈V has no solution) R1
OR
considering Cayley table for {V, ×10}
M1
the Cayley table is not a Latin square (or equivalent) R1
OR
considering cancellation law
eg, 5 ×10 9 = 5 ×10 1 = 5 M1
if {V, ×10} is a group the cancellation law gives 9 = 1 R1
OR
considering order of subgroups
eg, {1, 9} is a subgroup M1
it is not possible to have a subgroup of order 2 for a group of order 5 (Lagrange’s theorem) R1
THEN
so {V, ×10} is not a group AG
[2 marks]