# Generators and Multiplicative group $\mathbb{Z_7}$

In the multiplication table for $$\mathbb{Z^*_7}$$, each row has all the elements of the set

My questions are:

1. is each element a generator here? if not why?
2. is an element considered as generator only if $$Order(G) = Order(g)$$ i.e $$Order(g) = 6$$ hence $$g=3$$ is a generator.
3. is there any formula to find the number of generators?
4. Some website says phi(n) gives the order of group some says $$\varphi(n)$$ gives the number of generators some says $$\varphi(n)$$ gives the number of elements with a multiplicative inverse. Which of these is correct?

1. is each element a generator here? if not why?

Take the $$2$$ and calculate $$2 \cdot 2 \cdot 2 \equiv 8 \equiv 1 \mod7$$. Therefore, not every element is a generator.

1. is an element considered as generator only if $$Order(G) = Order(g)$$ i.e $$Order(g) = 6$$ hence $$g=3$$ is a generator.

A generator must generate all the element. If the order of $$Order(a) = x < g = Order(G)$$ then it cannot generates all the elements. $$a^x =1$$ and the cyclic sub-group generated by $$x$$ contains only $$x$$ elements:

$$a^1,a^2, \ldots, a^x = 1$$

1. is there any formula to find the number of generators?

No, there is no formula.

1. Some website says $$\varphi(n)$$ gives the order of group some says $$\varphi(n)$$ gives the number of generators some says $$\varphi(n)$$ gives the number of elements with a multiplicative inverse. Which of these is correct?

Remember the $$\mathbb{Z_7}$$ is a field.

• $$\varphi(n)$$ gives the order of the multiplicative group. In your case $$\varphi(7) = 7-1 = 6$$
• 2 and 4 are not generators therefore $$\varphi(n)$$ cannot give the number of generators.
• Being a field implies every element is invertible, thus $$\varphi(7)$$ gives the number of invertible elements, also for the composite case $$-\mathbb{Z_n}-$$, where $$n$$ is a composite number, $$\varphi(n)$$, gives the number of invertible elements.
• Take the 2 and calculate 2⋅2⋅2≡8≡1mod7. Therefore, not every element is a generator , could you explain this . why are we checking mod 7 of 8 ? – PDHide Dec 1 '18 at 17:36
• if you had checked the table i posted , it can be observed that each row recreates the elements of z7 , then why isnt each element not considered as a generator – PDHide Dec 1 '18 at 17:38
• Because we are talking about $\mathbb{Z_7}$. Each row contains distinct numbers since if there are more then one for a number say $a$ and $x$ is multiple times occurs then $x = a \cdot b= a \cdot c \Rightarrow b = c$. – kelalaka Dec 1 '18 at 17:42