7. (i) Define the order of a group, and the order of an element of a group.

(ii) Prove Lagrange's Theorem, that the order of a subgroup of a finite group G divides

the order of G.

(iii) The group table of the quaternion group of order 8 is given. Find the orders

of all elements of Q, and find all the right cosets of the subgroups H = {1, -1} and

J = {1,-1, j, -j}.

1.

1 -1

1

Ť

-1

1

2.

i

j

k -k

-k k

17

j-j -k

-i j -j

-2 j -j

i -j j -k

k -k

-k

k -1

1

k -k

j-j-i

j

i

1

k

k

i

j

-i

T

1

-k

-k

k

j

1

-1