# Isogeny to a semi-simple group

Let $$G$$ be a split reductive group over a field of characteristic zero. Is there always an isogeny $$G \to H \times T$$ with $$H$$ semi-simple and $$T$$ a split torus ?

I have in mind the case of $$\text{GL}_n$$ where we have (if I am not mistaken) the isogeny : $$\text{GL}_n \to \text{PGL}_n \times \mathbf{G}_m,\;\; g \mapsto (\overline{g},\det(g)).$$

Maybe it's even true that there is always a morphism $$G \to G^{ad} \times Z(G)$$ but I only need the less precise result I asked above.

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• Are you assuming that $G$ is connected? There is always a canonical isogeny to $G^{ad}\times G^{ab}$, where $G^{ab} = G/G^{der}$. – Keerthi Madapusi Pera Jan 11 at 19:27
• I am indeed assuming that $G$ is connected and your comment answers my question. Thanks a lot ! – user134482 Jan 12 at 17:31
• Note that the structure of a connected ereductive group may be rather complicated, when $G$ isn't semisimple or a torus: for examole, $D_4$ has a center of order 4 but exponent 2 and therefore can be combined with a torus in a complicated way. Even so, an isogeny of the sort mentioned by Pera is always possible. – Jim Humphreys Jan 12 at 17:33