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.