Groups of Order pq

M. Velrajan

Suppose p and q are primes with p < q.

i) If p ∤ q - 1, then G is cyclic, hence G ≅ ℤpq.

i.e. ℤpq is the only one group of order pq.

ii) If p | q - 1, then there exists a unique

non-abelian group and an abelian group of

order pq.

G = {aibj / 0 ≤ i < p and 0 ≤ j < q},

where ap = e = bq and aba-1 = br,

for some 1 < r < q and p is the smallest

positive integer such that rp ≡ 1(mod q)

and ℤpq are the only groups of order pq.

i.e. there are only two groups of order pq.

For proof