How to choose right and good notation and symbols?

Question

I learn group theory for research purpose. From last few time I am struggling to come with good notation and right symbols. Is there any good way to learn how to use best symbols.

For example suppose $g \in C_g$ is a right symbol or $x \in C_g$ is a good notation. Now suppose I have to show a group is normal in $G$ then should I use $xHx^{-1} =H$ or $gHg^{-1}$ which is better. Same question I have for generating set.

Question : How to choose right and good notation and symbols for research writing?

Answer 1

Two pieces of advice based on your examples:

• It often helps to use the same sort of letters for the same sort of things to speed up comprehension. e.g. $m$ and $n$ are always natural number or at least integers; $i$, $j$ and $k$ are indices. For groups, you usually see $g$ and then $h$ as group elements, which is why I slightly prefer $gHg^{-1}$ over $xHx^{-1}$.
• Always be aware of when you are using the same symbols for things. $g \in C_g$ means something entirely different to $x \in C_g$. The first is a summary of the proof that $g(g)g^{-1}=g$ and therefore $g$ must be in its own centraliser. The second just picks some element $x$ in the centraliser of g. If you write as though those notations mean the same thing you are going to confuse your reader.