Date | May 2014 | Marks available | 6 | Reference code | 14M.3srg.hl.TZ0.4 |
Level | HL only | Paper | Paper 3 Sets, relations and groups | Time zone | TZ0 |
Command term | Deduce and Prove that | Question number | 4 | Adapted from | N/A |
Question
Let be a homomorphism of finite groups.
Prove that , where is the identity element in and is the identity
element in .
(i) Prove that the kernel of , is closed under the group operation.
(ii) Deduce that is a subgroup of .
(i) Prove that for all .
(ii) Deduce that each left coset of K in G is also a right coset.
Markscheme
for M1A1
AG
[2 marks]
(i) closure: let and , then M1A1
A1
hence R1
(ii) K is non-empty because belongs to K R1
a closed non-empty subset of a finite group is a subgroup R1AG
[6 marks]
(i) M1
A1
A1
AG
(ii) clear definition of both left and right cosets, seen somewhere. A1
use of part (i) to show M1
similarly A1
hence AG
[6 marks]