First Sylow Theorem/Examples

Examples of Use of First Sylow Theorem

Alternating Group on $4$ Letters

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 $6 \divides 12$.