Should one really write $\text{det}A$, instead of $detA$, similarly $\text{Aut}G$ instead of $AutG$, or is $det$, $Aut$ acceptable?

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I would use \operatorname{det}(A), as that's what it is. –  gnometorule Feb 1 '13 at 4:51
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Note that $det(A)$ simply looks like it might mean $d\cdot e\cdot t(A)$. With respect to correct spacing in context, \operatorname is definitely the way to go.

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Better note that \det is predefined in LaTeX, so $\det(A)$ will give the correct result out of the box. –  egreg Feb 1 '13 at 9:52
@egreg Yes, \operatorname is rather for the generic case (such as Aut in the OP). –  Hagen von Eitzen Feb 1 '13 at 13:40
1. $det A$

2. $\text{det}A$

3. $\mathrm{det}A$

4. $\operatorname{det}A$

Personally, I prefer the second or third option or fourth option, as without italicization, it distinguishes the "operator" from the operand (matrix $A$ in this case), and/or fm other variables, which are italicized.

And the advantage of the last option is the slight increase in spacing between the operator ($\operatorname{det}$) and the operand.

See Deven's suggestion in the comment below: if you plan to use the "operator" a lot in a given post, you can write a "preamble" defining a "new command", which helps economize the effort involved in formulating a long post.

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\text should generally not be used: it uses the font of the surrounding text, which might be italic or small caps... It should be reserved for words that are part of the text. –  Mariano Suárez-Alvarez Feb 1 '13 at 6:12
As for defining operators, there is a standard \DeclareMathOperator command in amslatex.sty. –  Mariano Suárez-Alvarez Feb 1 '13 at 6:13
\operatorname not only increases the spacing when it is needed, it also removed the space when the next thing is a delimiter (like parentheses) and has a couple of other technical niceties that make it the correct option. –  Mariano Suárez-Alvarez Feb 1 '13 at 6:15
There already is \det, no need to define it yourself. –  morbusg Feb 1 '13 at 6:35
For things that are operators \DeclareMathOperator or \operatorname is the way to go, for example $\mathrm{sin}\theta$ just looks wrong. On the other hand, one might argue that things that are just names for sets, could just be written as \mathrm{Mat}(R,n)` (obviously I'd make a \Mat macro), because it would not be used in a context where something could be seen as multiplication. \text should never be used to get something appear in upright, as it will be italic in an italic context. –  daleif Feb 1 '13 at 9:11