Just a quick one. I've been sifting through many sources and found different definitions of rigidity when it comes to finite simple groups. I was hoping for some clarification.
http://www.maths.qmul.ac.uk/~raw/pubs_files/sgensweb.pdf
In this paper (under heading 3), the author defines a group to be rationally rigid if the "symmetrized structure constant" $\xi_G = 1$ (as defined page 3) and $g_1g_2g_3 = 1$.
On the other hand, if one looks at this question here: Possible repeated conjugacy classes in Thompson's Rigidity Criterion
we see that no mention is made of the symmetrized structure constant, only that if such a rigid tuple $(g_1,g_2,g_3)$ exists, then for any other tuple $(g^{'}_1,g^{'}_2,g^{'}_3)$ then there exists some unique $g$ such that the two tuples are conjugate.
Lastly, if we consider the following, Rigidity of conjugacy classes of finite centerless groups , Serre speaks of rigidity in terms of the group G acting transitively on some set $\Sigma$ with the desired properties of the group $G$.
I was just wondering whether these definitions were in fact equivalent. That is, for Thompson's Rigidity Criterion for the case of a triple generating system, would it be enough to have that a finite group $G$ is generated by $g_1,g_2,g_3$ such that $g_1g_2g_3=1$ with each $g_i$ $\in C_i$. I feel as if there's an extra condition that I am missing since each other three individual definitions impose a further condition.
Thank you for your help.