## Is the derived group of $G_2$ simply connected?

I am interested in conjugacy classes in connected reductive groups over a non-archimedean field $F$ of characteristic $0$, or its algebraic closure. On this topic it is often required that the group in question have simply connected derived group. Whether a group of type $G_2$ has simply connected derived group may be a well known fact, but I could not find a reference. That $G_2$ itself is simply connected is fairly easy to find, and I suspect its derived group is also. I would also be interested in any references.

Further, I am interested in whether the derived subgroups of other groups, for instance classical groups like $SL_n$, $SP_{2n}$, $SO_n$, have simply connected derived groups. References to characterizations of when a group has a simply connected derived group would also be much appreciated.

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Well, the groups of type $G_2$ are simple (or quasi-simple, I guess). Doesn't that help? – Mariano Suárez-Alvarez May 15 2012 at 22:18

This is basically a comment, but got too long. All the algebraic groups mentioned in the question are "simple" (also called "quasi-simple") over an algebraically closed field in the sense of having no closed connected normal subgroups except the entire group and the trivial group. Groups of type $G_2$ are in fact simple as abstract groups, but others might have a nontrivial finite center. For reductive groups there can be a torus in the center, as happens with $GL_n$; but here the derived group $SL_n$ is simply connected. On the other hand, $SO_n$ fails to be simply connected but is its own derived group. This kind of group structure for classical groups has been exposed in many books, though for the exceptional groups it's necessary to take a more unified viewpoint.

Concerning references, the basic structure and classification theory (going back to the 1956-58 Chevalley seminar) is worked out over an algebraically closed field in a number of textbooks with the same title Linear Algebraic Groups by Borel, Springer, and myself. Classifying "forms" over smaller fields is dealt with only partway in these books, while the scheme approach is found in SGA3, Demazure-Gabriel, and more recently the monograph Pseudo-reductive Groups by Conrad, Gabber, Prasad. In any case, it's important when working in this algebraic setting over fields which might not be of characteristic 0 to clarify the notion of "simply connected". Chevalley's notion, which agrees for the related complex Lie groups with the topological notion, depends just on a (split) maximal torus and its character group relative to the given reductive group.

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I assume that we are talking about algebraic groups. The groups SLn, SP2n, SOn, are all semisimple, hence equal to their derived groups, so you are asking whether the group itself is simply connected. This is true for the first two, but not SOn (it is covered by a Spin group). The simply connected almost-simple group of type G2 has trivial center. This means that every semisimple group of type G2 is simply connected. So if you have a reductive group whose derived group is of type G2, then that derived group is simply connected.

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