Show that the group U20 is not cyclic Show that the group U1

Show that the group U_20 is not cyclic. Show that the group U_18 is cyclic.

Solution

first we prove that the group U₂₀ is not cyclic.

The set of elements of U₂₀ is { 1, 3, 7 , 9 , 11 , 13 , 17 19 }

We can see that

< 1 > = { 1 }

< 3 > = { 3,9,7,1}

< 7 > = { 7,9,3,1 }

< 9 > = { 9,1}

<11> = { 11, 1}

< 13 > = { 13, 9 , 17 , 1 }

< 17 > = { 17,9,13,1}

< 19 > = { 19, 1 }

We can observe that < a > U₂₀ for every a U₂₀ .

Hence we proved the group U₂₀ is not cyclic.

Now we prove that the group U₁₈ is cyclic

The set of elements of U₁₈ is { 1 , 5 , 7 , 11, 13, 17 }

and we notice that 5¹ = 5 , 5² = 7, 5³ = 17 , 5⁴ = 13 , 5⁵ = 11, 5⁶ = 1

That is all the elements of U₁₈ are generated by 5 hence the group U₁₈ is cyclic