Group Isomorphism/Examples/Quotient Group of Z by 3Z with Quotient Group of A4 by K4

Example of Group Isomorphism
Let $\Z / 3 \Z$ denote the quotient group of the additive group of integers with the additive group of $3 \times$ the integers.

Let $A_4 / K_4$ denote the quotient group of the alternating group on 4 letters with the Klein $4$-group.

Then $\Z / 3 \Z$ is isomorphic to $A_4 / K_4$.

Proof
By Quotient Group of Integers by Multiples, $\Z / 3 \Z$ is the additive group of integers modulo $3$.

This is its Cayley table:

From Alternating Group on 4 Letters: Normality of Subgroups, $K_4$ is a normal subgroup of $A_4$.

We have that:
 * $\order {A_4 / K_4} = \index {A_4} {K_4} = \dfrac {12} 4 = 3$

As $3$ is prime, it follows from Prime Group is Cyclic that $A_4 / K_4$ and $\Z / 3 \Z$ are cyclic groups of order $3$.

The result follows from Cyclic Groups of Same Order are Isomorphic.

The Cayley table of $A_4$ makes this apparent: