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.
Dot notation for group extensions
abstract-algebraexact-sequencegroup-extensionsgroup-theorynotation
Related Solutions
In section 2, the paper gives this definition:
It looks to me that the summation notation means the sum for index $j$ (should be an integer) from $-(1+q)$ to $(1+q)$, and so $j=0$ should be included.
Best Answer
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 $12$.