First Sylow Theorem/Examples/Alternating Group on 4 Letters

Example of Use of First Sylow Theorem

The Alternating Group on 4 Letters $A_4$ is of order $12 = 2^2 \times 3$.

Thus the First Sylow Theorem tells us that $A_4$ has:

at least one subgroup of order $4$

at least one subgroup of order $2$

at least one subgroup of order $3$

These it has.

But it has no subgroup of order $6$, although $6 \divides 12$.

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.5$: Example $121$