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Date May 2008 Marks available 16 Reference code 08M.3srg.hl.TZ2.1
Level HL only Paper Paper 3 Sets, relations and groups Time zone TZ2
Command term Draw, Find, and Show that Question number 1 Adapted from N/A

Question

(a)     Draw the Cayley table for the set of integers G = {0, 1, 2, 3, 4, 5} under addition modulo 6, +6.

(b)     Show that {G, +6} is a group.

(c)     Find the order of each element.

(d)     Show that {G, +6} is cyclic and state its generators.

(e)     Find a subgroup with three elements. 

(f)     Find the other proper subgroups of {G, +6}.

Markscheme

(a)         A3

Note: Award A2 for 1 error, A1 for 2 errors and A0 for more than 2 errors.

 

[3 marks]

 

(b)     The table is closed     A1

Identity element is 0     A1

Each element has a unique inverse (0 appears exactly once in each row and column)     A1

Addition mod 6 is associative     A1

Hence {G, +6} forms a group     AG

[4 marks]

 

(c)     0 has order 1 (0 = 0),

1 has order 6 (1 + 1 + 1 + 1 + 1 + 1 = 0),

2 has order 3 (2 + 2 + 2 = 0),

3 has order 2 (3 + 3 = 0),

4 has order 3 (4 + 4 + 4 = 0),

5 has order 6 (5 + 5 + 5 + 5 + 5 + 5 = 0).     A3

Note: Award A2 for 1 error, A1 for 2 errors and A0 for more than 2 errors.

 

[3 marks]

 

(d)     Since 1 and 5 are of order 6 (the same as the order of the group) every element can be written as sums of either 1 or 5. Hence the group is cyclic.     R1

The generators are 1 and 5.     A1

[2 marks]

 

(e)     A subgroup of order 3 is ({0, 2, 4}, +6)     A2

Note: Award A1 if only {0, 2, 4} is seen.

 

[2 marks]

 

(f)     Other proper subgroups are ({0}+6), ({0, 3}+6)     A1A1

Note: Award A1 if only {0}, {0, 3} is seen.

 

[2 marks]

Total [16 marks]

Examiners report

The table was well done as was showing its group properties. The order of the elements in (b) was done well except for the order of 0 which was often not given. Finding the generators did not seem difficult but correctly stating the subgroups was not often done. The notion of a ‘proper’ subgroup is not well known.

Syllabus sections

Topic 8 - Option: Sets, relations and groups » 8.7 » The definition of a group {G,} .

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