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I came across the notation $A.B$ in many occassions while reading papers in Group Theory. However, I have not yet found any description of what this notation is supposed to mean. From context I suppose it means that if $G=A.B$, then $G$ is a solution to the extension problem $1\rightarrow A\rightarrow G\rightarrow B\rightarrow 1$, and the extension is not necessarily split. Is this the right way to interpret this symbol? In that case I suppose it would be useful in cases where the analysis does not depend on the particular extension.

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  • $\begingroup$ @DietrichBurde but there is no $G$ specified enveloping both $A$ and $B$ in general. I could take $G$ to be $A\rtimes B$ (assuming I have some action of $B$ on $A$) or $A\times B$ and $G=AB$ in both cases. My question is more specifically if there are any assumptions on the way you glue $A$ and $B$ together when you write $A.B$. $\endgroup$
    – mathma
    Dec 19, 2020 at 12:45

2 Answers 2

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This is known as ATLAS notation, and it was introduced and widely used in the ATLAS of Finite Groups. It has become a standard notation among (particularly finite) group theorists, and it is very concise and useful, but it should be used with care.

In fact you have guessed the meaning correctly. If $A$ and $B$ are groups (or isomorphism types of groups), then $A.B$ means any group with normal subgroup isomorphic to $A$ with corresponding quotient group isomorphic to $B$.

There are refinements. For example $A:B$ denotes a split extension, and you would normally use $A \times B$ if you knew that it was a direct product, but just writing $A.B$ does not imply that it is not split or a direct product, it probably just means that you do not know or care.

Other conventions include using a number to denote a cyclic group of that order (as in $3.A_6$or $A_5.2 = S_5$), or a number like $[12]$ in square brackets to denote an unspecified group of order $12$.

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  • $\begingroup$ Thank you very much @Derek Holt for your answer. Indeed, I had also seen the : symbol and knowing its meaning is also really helpful. $\endgroup$
    – mathma
    Dec 19, 2020 at 13:15
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In addition to Derek Holt's answer I am adding here a reference to a webpage by Tim Dokchitser that explains the ATLAS notation

https://people.maths.bris.ac.uk/~matyd/GroupNames/extensions.html

Edit: Thanks to Derek Holt who clarified the difference between the notation $A.B$ in the ATLAS and the page above. In the ATLAS $A.B$ is an arbitrary extension, whereas in the webpage it denotes a non-split extension.

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    $\begingroup$ Unfortunately, what it says on that webpage is not an accurate description of ATLAS notation. It says there that $N.Q$ denotes a non-split extension, whereas in fact it denotes an arbitrary extension. There is a notation $N ^{\large \cdot} Q$ that specifically denotes a non-split extension. $\endgroup$
    – Derek Holt
    Aug 11, 2021 at 11:00

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