Group theory and generalizations Example

The alternating group A5 is simple

Builds on Simplicity of the alternating group Conjugate elements Lagrange's theorem

Example

The group $A_5$ has order $60$ . Its conjugacy classes have sizes $1$ , $15$ , $20$ , $12$ and $12$ . The class of size $1$ is $\{e\}$ , and the class of size $20$ contains the $3$ -cycles. The class of size $15$ contains the products of two disjoint transpositions. The $24$ five-cycles split into two classes of size $12$ . A normal subgroup $N$ of $A_5$ is a union of conjugacy classes, and it contains the class $\{e\}$ . By Lagrange's theorem, $|N|$ divides $60$ . No sum of $1$ and some of the numbers $15, 20, 12, 12$ divides $60$ , except $1$ and $60$ . Thus $N = \{e\}$ or $N = A_5$ , and $A_5$ is simple. It is the smallest simple group that is not abelian.

Reading path

To understand this page from the ground up, read these 57 pages in order. The foundations come first. The path takes the proof Proof of Lagrange's theorem.

Cycle notation concept
Element definition
Group concept
Left coset concept
Proposition concept
Set definition
Conjugate elements definition
Group homomorphism definition
Subgroup definition
Symmetric group definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Empty set object
Family of sets definition
Ordered pair definition
Conjugate subgroup definition
Disjoint cycles commute statement
Intersection definition
Kernel of a homomorphism definition
Transposition definition
Union definition
Left coset definition
Right coset definition
Subgroup criterion statement
Set-builder notation concept
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Disjoint sets definition
Decomposition of a permutation into disjoint cycles statement
Normal subgroup definition
Subset definition
Intersection of subgroups is a subgroup statement
Union of a family of sets definition
Cosets have the same cardinality as the subgroup statement
Binary relation definition
Partition of a set definition
Subgroup generated by a subset definition
Decomposition of a permutation into transpositions statement
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Sign of a permutation definition
Equivalence relation definition
Sign is a homomorphism statement
Even and odd permutations definition
Sign of a cycle statement
Alternating group definition
Equivalence class definition
Equivalence classes form a partition statement
The alternating group is generated by three-cycles statement
Simplicity of the alternating group statement
Proof of Lagrange's theorem proof
Lagrange's theorem statement
The alternating group A5 is simple example

Tag	`0GKQKF0FXJ`
URI	`m2://0GKQKF0FXJ`
Kind	example
Writing	W0 stub
MSC	20B30, 20D06
Instance of	Alternating group
Uses	Simplicity of the alternating group, Conjugate elements, Lagrange's theorem

The alternating group A5 is simple

Example

Reading path

Further reading

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