I'm interested in the determination of a group $\operatorname{Aut}( \operatorname{GL}_n \mathbb C)$.

A class of examples of automorphisms of $\operatorname{GL}_n \mathbb C$ is given by conjugations $c_g$, for $g \in \operatorname{GL}_n \mathbb C$: $$ c_g(A) = gAg^{-1}.$$

These automorphisms form a normal subgroup of $\operatorname{Aut}( \operatorname{GL}_n \mathbb C)$, a group of inner automorphisms.

But for $n > 2$, we have an example of an outer automorphism: define $$ \iota(A) = (A^T)^{-1} = (A^{-1})^T.$$

Why we can't have $\iota = c_g$, for some $g \in G$? Take $\lambda \in \mathbb C$ such that $\lambda^n = 1, \lambda^2 \neq 1$. Then: $$\iota(\lambda I) = \lambda^{-1} I,$$

(which isn't equal to $\lambda I$ because of $\lambda^2 \neq 1$)


$$ c_g(\lambda I) = \lambda I,$$

for all $g$. My questions are, in an order of increasing difficulty:

1) Is $\operatorname{Aut}( \operatorname{GL}_2 \mathbb C) = \operatorname{Inn}( \operatorname{GL}_2 \mathbb C)$?

2) Is $ \operatorname{Aut}( \operatorname{GL}_n \mathbb C)/ \operatorname{Inn}( \operatorname{GL}_n \mathbb C) = \{\mathrm{id}, \iota \} \simeq \mathbb Z / 2\mathbb Z $?

3) What is $\operatorname{Aut}( \operatorname{GL}_n \mathbb C)$?

I would be really grateful if someone could give me a good reference in which all of these questions are discussed (especially if it's done for general fields).

  • 2
    $\begingroup$ Complex conjugation is an example of a non-inner automorphism of $GL_n(\mathbb C)$. This shows that the answer to 1) is negative. $\endgroup$ – Ariyan Javanpeykar Jan 17 '17 at 13:14
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    $\begingroup$ For one thing, there is also the automorphism coming from complex conjugation. $\endgroup$ – Tobias Kildetoft Jan 17 '17 at 13:14
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    $\begingroup$ The Hermitian conjugate, while not an automorphism (it reverses multiplication, like transposes), becomes one if you, like with $\iota$, compose with the inverse. $\endgroup$ – Arthur Jan 17 '17 at 13:16
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    $\begingroup$ Note that every automorphism of $\mathbf C$, like complex conjugation, extends to an automorphism of $\mathbf{GL_n} \mathbf C$ that is not inner. And $\mathbf C$ has a lot of "wild" automorphisms as discussed, for example, here $\endgroup$ – Alex Macedo Jan 17 '17 at 13:31
  • $\begingroup$ @Arthur Now I feel really silly for writing down $\iota$ this way and not seeing I could do the same thing for Hermitian conjugate. :D $\endgroup$ – ante.ceperic Jan 17 '17 at 13:35

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