Alternating Group on 4 Letters/Subgroups/Examples/Order 3

Subgroups of the Alternating Group on $4$ Letters
Let $A_4$ denote the alternating group on $4$ letters, whose Cayley table is given as:

Let $P$ denote the subset of $A_4$:
 * $P := \set {e, a, p}$

Then $P$ is a subgroup of $A_4$.

Its left cosets are:

Its right cosets are:

Proof
We have that:

Thus $\set {e, a, p}$ forms a cyclic group generated by $a$.

Thus $\set {e, a, p}$ is a group which is a subset of $A_4$.

Hence by definition $\set {e, a, p}$ is a subgroup of $A_4$.

Then:

and: