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YCor
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Injection of $G(k)/Z(k)$ into the $(G/Z)(k)$

In the first answer to the linked question it is mentioned that "the isogeny $G\to G^{ad}$ induces an injection of groups $G(k)/Z(k)\to G^{ad}(k)$". Is there a reference for this result? Carter's book "Simple groups of Lie types""Simple groups of Lie type" is mentioned but there is nothing there about it as far as I can find.

Injection of $G(k)/Z(k)$ into the $(G/Z)(k)$

In the first answer to the linked question it is mentioned that "the isogeny $G\to G^{ad}$ induces an injection of groups $G(k)/Z(k)\to G^{ad}(k)$". Is there a reference for this result? Carter's book "Simple groups of Lie types" is mentioned but there is nothing there about it as far as I can find.

Injection of $G(k)/Z(k)$ into $(G/Z)(k)$

In the first answer to the linked question it is mentioned that "the isogeny $G\to G^{ad}$ induces an injection of groups $G(k)/Z(k)\to G^{ad}(k)$". Is there a reference for this result? Carter's book "Simple groups of Lie type" is mentioned but there is nothing there about it as far as I can find.

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Μάρκος Καραμέρης
Μάρκος Καραμέρης
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In the first answer to the linked questionlinked question it is mentioned that "the isogeny $G\to G^{ad}$ induces an injection of groups $G(k)/Z(k)\to G^{ad}(k)$". Is there a reference for this result? Carter's book "Simple groups of Lie types" is mentioned but there is nothing there about it as far as I can find.

In the first answer to the linked question it is mentioned that "the isogeny $G\to G^{ad}$ induces an injection of groups $G(k)/Z(k)\to G^{ad}(k)$". Is there a reference for this result? Carter's book "Simple groups of Lie types" is mentioned but there is nothing there about it as far as I can find.

In the first answer to the linked question it is mentioned that "the isogeny $G\to G^{ad}$ induces an injection of groups $G(k)/Z(k)\to G^{ad}(k)$". Is there a reference for this result? Carter's book "Simple groups of Lie types" is mentioned but there is nothing there about it as far as I can find.

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Μάρκος Καραμέρης
Μάρκος Καραμέρης
  • 131
  • 4

Injection of $G(k)/Z(k)$ into the $(G/Z)(k)$

In the first answer to the linked question it is mentioned that "the isogeny $G\to G^{ad}$ induces an injection of groups $G(k)/Z(k)\to G^{ad}(k)$". Is there a reference for this result? Carter's book "Simple groups of Lie types" is mentioned but there is nothing there about it as far as I can find.