Let $a,b\in \mathbb{F}_p^*$ with orders $o_p(a)=|\langle a \rangle|=\alpha$ and $o_p(b)=|\langle b \rangle|=\beta$. I have few questions:
1) Is it true in this case ($\mathbb{F}_p^*$ cyclic) that $o_p(a,b)=|\langle a,b \rangle| = \rm{lcm}(\alpha,\beta)$? I tried to prove it (supposing it's true!) using primitive roots, i.e. saying that $a=g^k$ and $b=g^h$ for some fixed primitive root $g$, but I got stuck.
2) Does the the previous relation also holds in generic abelian group $G$?
3) Is there a formula for $o_p(a,b,c)=|\langle a,b,c \bmod p\rangle|$?
Thanks in advance!
1) Yes. This holds in any cyclic group. The subgroup $\langle a,b\rangle$ is cyclic as a subgroup of a cyclic group. Use the uniqueness of a subgroup of a given order to get the claim. Your idea of writing everything in terms of the powers of a primitive root also works. The subgroup generated by $g^h$ and $g^k$ is generated by $g^{\gcd(h,k,p-1)}$.
2) No. If $a,b$ are distinct elements of the Klein 4-group, then $\langle a,b\rangle$ has order four, but the lcm is $=2$.
3) Iterating the result from part 1) you get that the order of this group is the l.c.m. of the orders of the three generators.