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Date November 2009 Marks available 9 Reference code 09N.3srg.hl.TZ0.5
Level HL only Paper Paper 3 Sets, relations and groups Time zone TZ0
Command term Show that Question number 5 Adapted from N/A

Question

Let {G , } be a finite group of order n and let H be a non-empty subset of G .

(a)     Show that any element hH has order smaller than or equal to n .

(b)     If H is closed under , show that {H , } is a subgroup of {G , }.

Markscheme

(a)     if hH then hG     R1

hence, (by Lagrange) the order of h exactly divides n

and so the order of h is smaller than or equal to n     R2

[3 marks]

 

(b)     the associativity in G ensures associativity in H     R1

(closure within H is given)

as H is non-empty there exists an hH , let the order of h be m then hm=e and as H is closed eH     R2

it follows from the earlier result that hhm1=hm1h=e     R1

thus, the inverse of h is hm1 which H     R1

the four axioms are satisfied showing that {H , } is a subgroup     R1

[6 marks]

Total [9 marks]

Examiners report

Solutions to this question were extremely disappointing. This property of subgroups is mentioned specifically in the Guide and yet most candidates were unable to make much progress in (b) and even solutions to (a) were often unconvincing.

Syllabus sections

Topic 8 - Option: Sets, relations and groups » 8.11 » Subgroups, proper subgroups.

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