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I am reading an article on elementary homological algebra and have a trouble understanding one statement. Let $R$ be a ring and $A,B,C$ modules over $R$. Let $S$ be a set of exact sequences of the form $$ 0\rightarrow A\rightarrow B\rightarrow C \rightarrow 0 $$ The article says

$\operatorname{Aut}(B)$ acts on $S$ with stabilizer $1+\alpha \operatorname{Hom}(C,A)\beta$ where $\alpha,\beta$ are the maps fitting in the short exact sequence of the trivial extension $$ 0 \rightarrow A\stackrel{\alpha}{\rightarrow} A\oplus C \stackrel{\beta}{\rightarrow} C \rightarrow 0 $$

Firstly I don't quite understand what $1+\alpha \operatorname{Hom}(C,A)\beta$ means (what is $1+\dots$?) and secondly don't see why the stabilizer of $\operatorname{Aut}(B)$ is identified with the above set.

Could anyone kindly explain what is going on? Thank you very much.

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    Please, try to make the title of your question more informative. E.g., *Why does $a imply $a+c?* is much more useful for other users than *A question about inequality.* From [How can I ask a good question?](http://meta.math.stackexchange.com/a/589/): *Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader.*2012-11-25
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    Sorry about that. I initially tried to give better title, but could not make it short and precise.2012-11-26

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