1. Cyclic group

Let $\displaystyle G$ denote a group and let $\displaystyle g \in G$. In each case show $\displaystyle G = < g >$.

a. $\displaystyle |G| = 12, g^4 \neq 1, g^6 \neq 1$.
b. $\displaystyle |G| = 40, g^8 \neq 1, g^{20} \neq 1$.
c. $\displaystyle |G| = 12, g^{30} \neq 1, g^{20} \neq 1, g^{12} \neq 1$.
d. Generalize.

2. Originally Posted by chiph588@
Let $\displaystyle G$ denote a group and let $\displaystyle g \in G$. In each case show $\displaystyle G = < g >$.

a. $\displaystyle |G| = 12, g^4 \neq 1, g^6 \neq 1$.
b. $\displaystyle |G| = 40, g^8 \neq 1, g^{20} \neq 1$.
c. $\displaystyle |G| = 12{\color{red}0}, g^{30} \neq 1, g^{20} \neq 1, g^{12} \neq 1$.
d. Generalize.
The order of an element of a group always divides the order of the group (so in example c., |G| can't be 12, it must be 120).

In example a., the order of g must be 1, 2, 3, 4, 6 or 12 (those being the divisors of 12. If g has order 1 then g is the identity and so g to any other power is also 1. If g has order 2 then $\displaystyle g^4 = 1$. If g has order 3 then $\displaystyle g^6=1$. So if $\displaystyle g^4\ne1$ and $\displaystyle g^6\ne1$ then the only remaining possibility is that g has order 12. Thus g generates the whole of G.

The other two examples are similar. I'll leave you to think about the generalisation.