Exercise 1699Algebra › GroupsOlympiadAdded 3 Nov 2026

Cauchy's theorem applied to $A_4$

#Cauchy#alternating group#elements of order

Statement

Using Cauchy's theorem, show that $A_4$ has elements of order $2$ and order $3$ .

Hint

$|A_4|=12=2^2\cdot3$ . Cauchy's theorem gives an element of each prime order.

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Order $3$ . The $3$ -cycles $(123),(124),(134),(234)$ and their inverses are $8$ elements of order $3$ .

Order $2$ . Products of disjoint transpositions like $(12)(34)$ have order $2$ . There are $3$ such elements: $(12)(34),(13)(24),(14)(23)$ .

Cauchy. Since $|A_4|=12$ is divisible by $2$ and $3$ , Cauchy's theorem guarantees elements of order $2$ and $3$ .

Conclusion. $A_4$ has $8$ elements of order $3$ and $3$ elements of order $2$ , plus the identity.