Give the answer to each part there are 3 parts Prove that th

Give the answer to each part (there are 3 parts)

Prove that the property that a group is cyclic is a structural property

Solution

1. Since structural property says that elements of a group are occupied with operations that converts one element of a group to another Or two elements combine to form third element.

For example: As we know that Z₁₁ is a cyclic group. Therefore addition of any two elements such as (5+9) mod 11 gives you 3 as a result which is the element of same group. (7)^-1 in the same group is 8. And also multilication on any 2 elements such as 5*3 mod 11 will give you 4 which is the element of same group.

Therefore we can say cyclic group follow structural property.

2. Let G has no proper non-trivial sub-groups.

Now take an element g in G for which g is not equal to e (identity). Assume the cyclic sub-group <g>. This subgroup contains atleast e and g, so it is not trivial. But G has no proper sub-groups, therefore it should be <g>=G. Thus G is cyclic by the definition of Cyclic group.

3.You are given U_n = {z C : zⁿ = 1}

z⁶ = 1 = e⁰

z⁶ = e^i2l

z = e^i2l/6 =e^il/3 where l=0,1,2,3.....

and we know that e^i2l = 1,

lZ is used.