Group theory and generalizations Example

Groups of order pq

Builds on Third Sylow theorem The number of Sylow subgroups divides the index Internal direct product Product of cyclic groups is cyclic iff the orders are coprime

Example

Let $p < q$ be prime numbers such that $p$ does not divide $q - 1$ , and let $G$ be a group of order $pq$ . The number $n_q$ divides $p$ and satisfies $n_q \equiv 1 \pmod q$ , so $n_q = 1$ , because $p < q$ . The number $n_p$ divides $q$ and satisfies $n_p \equiv 1 \pmod p$ ; since $p \nmid q - 1$ , $n_p = 1$ . So $G$ has a normal subgroup $P$ of order $p$ and a normal subgroup $Q$ of order $q$ . They meet only in $e$ , so $G$ is the internal direct product of $P$ and $Q$ . Thus $G \cong \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/q\mathbb{Z} \cong \mathbb{Z}/pq\mathbb{Z}$ , and $G$ is cyclic. For example, each group of order $15$ is cyclic. If $p$ divides $q - 1$ , there is also a group of order $pq$ that is not cyclic, for example the dihedral group $D_3$ of order $6$ .

Reading path

To understand this page from the ground up, read these 77 pages in order. The foundations come first. The path takes the proof Proof of Lagrange's theorem.

Element definition
Group concept
Left coset concept
Proposition concept
Set definition
Conjugate elements definition
Direct product of groups definition
Group homomorphism definition
Laws of exponents in a group statement
Order of a group definition
Subgroup definition
Symmetric group definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Empty set object
Family of sets definition
Ordered pair definition
Order of an element definition
Intersection definition
Union definition
Conjugate subgroup definition
Group action definition
Cyclic subgroup generated by an element definition
Set-builder notation concept
Left coset definition
Product of two subgroups definition
Right coset definition
Subgroup criterion statement
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Disjoint sets definition
P-group definition
g to the n equals 1 iff the order divides n statement
Normal subgroup definition
Normaliser definition
Subset definition
Stabiliser definition
Union of a family of sets definition
Orbit definition
Cyclic group definition
Intersection of subgroups is a subgroup statement
Index of a subgroup definition
Cosets have the same cardinality as the subgroup statement
Binary relation definition
Sylow p-subgroup definition
Order of a cyclic subgroup equals the order of its generator statement
Partition of a set definition
Internal direct product definition
Normal core definition
Fixed points of an action definition
Orbits partition the set statement
Transitive action definition
Order of a group is the index times the order of the subgroup statement
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Cyclic groups of the same order are isomorphic statement
Action on left cosets example
Equivalence relation definition
Product of cyclic groups is cyclic iff the orders are coprime statement
Orbit-stabiliser theorem statement
Equivalence class definition
Equivalence classes form a partition statement
Proof of Lagrange's theorem proof
Lagrange's theorem statement
Size of an orbit times the size of the stabiliser statement
Fixed points of a p-group action statement
Third Sylow theorem statement
Cauchy's theorem statement
First Sylow theorem statement
Second Sylow theorem statement
The number of Sylow subgroups is the index of the normaliser statement
The number of Sylow subgroups divides the index statement
Groups of order pq example

Tag	`0YBR6QPY4Y`
URI	`m2://0YBR6QPY4Y`
Kind	example
Writing	W0 stub
MSC	20D20
Uses	Third Sylow theorem, The number of Sylow subgroups divides the index, Internal direct product, Product of cyclic groups is cyclic iff the orders are coprime

Groups of order pq

Example

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