Question

Show that the additive group $Z_2\times Z_3$ is cyclic.

Short answer

It is proved that Z₂x Z₃ is cyclic.

Step-by-step solution

Definition of a cyclic group

A cyclic group G is a group in which there is at least one element a such as $\mathrm{a}\in \mathrm{G}$so that {a} = {${\mathrm{a}}^{\mathrm{n}}|\mathrm{a}\in \mathrm{Z}$} is a subgroup of G.

Proving that Z2 x Z3 is indeed cyclic

The two groups are:

Z₂ = {0,1}

Z₃= {0,1,2}

Thus Z₂ x Z₃is:

Z₂ x Z₃= {(0,0), (0,1), (0,2), (1,0), (1,1), (1,2)}

Now, if we take successive multiples of (1,1) ,we get,

(1,1), (0,2), (1,0), (0,1), (1,2), (0,0)

Since_1_is a generator for both Z₂ and Z₃, let’s consider the powers of $(1,1)\in Z_2\times Z_3(1,1)$:

$\left\{n\right(1,1\left)\right|n\in Z\}$=$\left\{\right(0,0),(1,1),(0,2),(1,0),(0,1),(1,2\left)\right\}=Z_2\times Z_3$

So, (1,1) is a generator of Z₂ x Z₃and it is cyclic.

Hence, it is proved that_Z₂ x Z_{3_}is cyclic.