Groups of Order pq: When Two Primes Collide

After mastering groups of prime order, students ask me: "What happens when we multiply two primes?" This is where group theory becomes really interesting – we get multiple possibilities!


Let p and q be distinct primes with p < q.

We want to classify all groups of order n = pq.

The Main Result

Theorem: There always exists at least one group of order pq, namely the cyclic group Z_{pq}.

Additional groups exist depending on whether q ≡ 1 (mod p).

Case 1: If q ≢ 1 (mod p)

Only one group: Z_{pq} (cyclic, abelian)

Case 2: If q ≡ 1 (mod p)

Two groups:


Z_{pq} (cyclic, abelian)

One non-abelian group of order pq


Why Does This Happen?

The key is Sylow Theory and the existence of normal subgroups.

Fact: Every group of order pq has:

A subgroup of order p (by Cauchy's theorem)

A subgroup of order q (by Cauchy's theorem)


The question is: are these subgroups normal?

Result: The subgroup of order q is ALWAYS normal.

The subgroup of order p is normal if and only if q ≢ 1 (mod p).


Examples

Example 1: Order 15 (p=3, q=5)

Check: 5 ≡ 2 (mod 3), so 5 ≢ 1 (mod 3)

Only one group: Z₁₅

This group is cyclic and abelian.


Element 1 generates: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}

Element 2 also generates the whole group

Number of generators: φ(15) = 8


Example 2: Order 6 (p=2, q=3)

Check: 3 ≡ 1 (mod 2), so q ≡ 1 (mod p)

Two groups exist:

Z₆: Cyclic, abelian

Elements: {0, 1, 2, 3, 4, 5}

Generators: {1, 5}


S₃: Symmetric group, non-abelian

Elements: {e, (12), (13), (23), (123), (132)}

NOT cyclic

Example of non-commutativity: (12)(123) = (13) but (123)(12) = (23)


Example 3: Order 35 (p=5, q=7)

Check: 7 ≡ 2 (mod 5), so 7 ≢ 1 (mod 5)

Only one group: Z₃₅

All groups of order 35 are isomorphic to Z₃₅.


Example 4: Order 10 (p=2, q=5)

Check: 5 ≡ 1 (mod 2), so q ≡ 1 (mod p)

Two groups exist:

Z₁₀: Cyclic, abelian

D₅: Dihedral group (symmetries of regular pentagon), non-abelian



Test for Cyclicity: Does the group have an element of order pq?

If YES: Group is Z_{pq}

If NO: Group is the non-abelian type


Test for Abelian: Do all elements commute?

If YES: Must be Z_{pq}

If NO: Must be the non-abelian group


Subgroup Structure

For Z_{pq} (Cyclic)

By Lagrange, subgroup orders divide pq: {1, p, q, pq}


One subgroup of order 1: {e}

One subgroup of order p

One subgroup of order q

One subgroup of order pq: the whole group


Total: 4 subgroups

For Non-Abelian Groups


One subgroup of order 1: {e}

p subgroups of order q (all conjugate)

One subgroup of order p (normal)

One subgroup of order pq: the whole group



The Cyclic Condition

Theorem: Z_{pq} exists and is cyclic when gcd(p, q) = 1.

Since p and q are distinct primes, gcd(p,q) = 1 always holds.

By the Chinese Remainder Theorem: Z_{pq} ≅ Z_p × Z_q

This is why Z₆ ≅ Z₂ × Z₃, and Z₁₅ ≅ Z₃ × Z₅.


Multiple Choice Quiz

Question: Let G be a non-abelian group of order 14. Which is TRUE?

A) G is cyclic

B) G ≅ Z₁₄

C) G has 7 subgroups of order 2

D) G ≅ D₇

Answer: D) G ≅ D₇

Explanation:


A) False – cyclic groups are abelian

B) False – Z₁₄ is abelian

C) False – G has one subgroup of order 2 (normal)

D) True – the non-abelian group of order 14 is D₇


Why This Classification Matters

In Cryptography: Understanding group structure helps analyze security.

In Galois Theory: Groups of order pq appear as Galois groups.

In Representation Theory: Different structures have different representations.

In Chemistry: Molecular symmetries often have orders that are products of primes.

The Beauty of the Result

What I love about this theorem is how one simple congruence condition (q ≡ 1 mod p) determines whether we get one or two groups!

Mathematics reveals its elegance through such clean classifications. The interplay between number theory (congruences) and group theory (structure) is beautiful.

For more problems on classifying groups and understanding their structure, visit my YouTube channel "Maths  mastery with Dr. Upasana P Taneja" where I work through many examples of groups with various orders!

Understanding groups of order pq is a crucial step toward classifying all finite groups. It shows how combining primes creates richer structure than individual primes alone.


Two primes, finite possibilities – that's the power of classification!

Dr. Upasana Pahuja Taneja

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