Let
G be a finite group and H a subgroup of G. Let a be an element of G
and aH = {ah : h is an element of H} be a left coset of H. If b is
an element of G as well and the intersection of aH bH is non-empty
then aH and bH contain the same number of elements in G. Thus
conclude that the number of elements in H, o(H), divides the number
of elements...
Prove that in a finite cyclic group, each subgroup has size dividing the size of the group. Conversely, given a positive divisor of the size of the group, there is a subgroup of that size.
Let a be an element of a finite group G. The order of a is the
least power k such that ak = e.
Find the orders of following elements in S5
a. (1 2 3 )
b. (1 3 2 4)
c. (2 3) (1 4)
d. (1 2) (3 5 4)