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@user47958: The first one. This is a generalization of the notion of exponent. That is, every group of exponent dividing $n$ is of generalized exponent $n$.
So maybe a question as follows is interesting: Is there a function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that there is no finite group $G$ of order coprime to 3 and size greater than $f(n)$ and no field $\mathbb{F}$ such that the group algebra $\mathbb{F}[G]$ contains elements $\alpha$ and $\beta$ with the following properties: $\alpha \beta=0$, $|supp(\alpha)|=3$, $|supp(\beta)|=n$?
@AndreasThom: Another motivation to propose the question is the question of the existence of zero divisors with support of size 3 in the group algebra of torsion-free groups which are residually finite. The number ``3" is the first unsettled.
Many thanks. This perfectly and very simply answer my question. Actually the answer shows all elements in the augmentation ideal of the group algebra whose support is of size 3 can be chosen as a candidate for the answer over any field with more than 2 elements. The construction may interpret as canonical as it is valid over any finite group and any field with more than 2 elements. Can one find a zero divisor with support of size $3$ over $GF(2)$? anyway again many thanks for your answer.
@DerekHolt: Yes. This answers negatively my question: You take the graph $\Gamma$ as follows $1 \overset{a}{\longrightarrow} 2$, $1 \overset{b}{\longrightarrow} 2$, $1 \overset{c}{\longrightarrow} 2$, $1 \overset{d}{\longrightarrow} 2$. A graph with two vertices $1,2$ and four directed edges between these two vertices. Are there examples in which we have no multiple edges between vertices? It depends to how one choose the graph $\Gamma$. If one consider the graph $\Gamma$ with three disjoint cylces, then the answer to my question is positive. Anyway, thanks for your try to answer my question
@DerekHolt: What I mean by the graph $\Gamma$: it is a directed labelled graph in which we have at least one cycle corresponding to each relation $r_i$. The graph $\Gamma$ is not unique: e.g., one may consider $m$ disjoint cycles each of them corresponds to a relation $r_i$; or one may glue properly the cycles to have a graph with less number of connected components. May be referring to the Van Kampen diagrams is not necessary and confusing; I am trying to draw such a graph $\Gamma$ for an specific example but at the moment I do not know how to do it in MathJax...
