
Group theory
A B C D F G H J K L M N
A C G J M B K A D N F H L
B F H L K A M B N D C G J
C J K D H G N C M L B A F
D M L H C N F D A B K J G
F L M N G H D F K J A B C
G B A F N C H G L M J K D
H A B C D F G H J K L M N
J D N M A K L J H F G C B
K G C B L J A K F H D N M
L N D K B M J L G C H F A
M H F A J L B M C G N D K
N K J G F D C N B A M L H
a) Findthe identity
b) Find the inverse of each element
c) Find the order ofeach of A, Jand N
d) State with reasons which of the elements are reflections andwhich are rotations. For the ones which are rotations describe which rotationis represented by which element. For the ones which are reflections you are notasked to give any details. Note there is more than one correct answer for thisquestion. For a bonus mark, explain why there is more than one answer.
e)State the possibleorders of the proper subgroups.
f) Find a subgroup of each of the possible (proper) orders anddraw up their group tables.
g) For each subgroup decide if it is cyclic and /or Abelian
h) For the subgroup of order 3, find its left cosets and itsright cosets. Is the subgroup normal?
i) Find a subgroup of index 3. Give its left cosets and itsright cosets. Is the subgroup normal?
j) Ifyou found that either or both of your subgroups in parts h) and i) were normal,state the members of the associated quotient group and draw up its group table.
Ive completed parts of the question but i am stuck for the most part, any help would be hugley appreciated. thank you.