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

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Example of Group Isomorphism

Let $\Z / 3 \Z$ denote the quotient group of the additive group of integers by the additive group of $3 \times$ the integers.

Let $A_4 / K_4$ denote the quotient group of the alternating group on 4 letters by 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:

$\begin{array}{r|rrr} \struct {\Z_3, +_3} & \eqclass 0 3 & \eqclass 1 3 & \eqclass 2 3 \\ \hline \eqclass 0 3 & \eqclass 0 3 & \eqclass 1 3 & \eqclass 2 3 \\ \eqclass 1 3 & \eqclass 1 3 & \eqclass 2 3 & \eqclass 0 0 \\ \eqclass 2 3 & \eqclass 2 3 & \eqclass 0 3 & \eqclass 1 3 \\ \end{array}$


From Normality of Subgroups of Alternating Group on 4 Letters, $K_4$ 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:

$\begin{array}{c|cccc|cccc|cccc} \circ & e & t & u & v & a & b & c & d & p & q & r & s \\ \hline e & e & t & u & v & a & b & c & d & p & q & r & s \\ t & t & e & v & u & b & a & d & c & q & p & s & r \\ u & u & v & e & t & c & d & a & b & r & s & p & q \\ v & v & u & t & e & d & c & b & a & s & r & q & p \\ \hline a & a & c & d & b & p & r & s & q & e & u & v & t \\ b & b & d & c & a & q & s & r & p & t & v & u & e \\ c & c & a & b & d & r & p & q & s & u & e & t & v \\ d & d & b & a & c & s & q & p & r & v & t & e & u \\ \hline p & p & s & q & r & e & v & t & u & a & d & b & c \\ q & q & r & p & s & t & u & e & v & b & c & a & d \\ r & r & q & s & p & u & t & v & e & c & b & d & a \\ s & s & p & r & q & v & e & u & t & d & a & c & b \\ \end{array}$


$\blacksquare$


Sources

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