There is a conclusion on my textbook:
All semisimple algebraic groups of type $F_4$ are isomorphic.
I was confused because of the fundamental groups.
Is it true that two semisimple algebraic groups could be isomorphic only when their fundamental groups are isomorphic? So, all semisimple algebraic groups of type $F_4$ have isomorphic fundamental groups?
But if $\Lambda$ denotes the weight lattice, and $\Lambda_r$ denotes the root lattice, then under calculation, I found that $[\Lambda:\Lambda_r]=29$, so the fundamental group may have order $1$ or $29$... Am I wrong?
Moreover, is there any connection between the fundamental group (of the semisimple algebraic group) and the diagram automorphism of the Dynkin diagram of the corresponding root system? The semisimple algebraic groups of type $F_4$ have only one fundamental group (up to isomorphism), because the Dynkin diagram of $F_4$ has no nontrivial diagram automorphism? Is this true? And if it is, why?
Thank you very much.
[I think this has a deep connection to Lie algebras, so I will give two tags to this question.]