assume $ G $ is a group and $ S subassembly G $ is his finite subset. Let's define the *commuting graph* of $ G $ in respect of $ S $ as $ Comm (G, S) $ – a graph, where the vertices are the elements of $ S $, and they are connected with edges if they commute.

Let's define the *universality of displacements* of $ G $ as $ UGT (G) $ – the maximum $ k in mathbb {N} $ such as sny graph with k $ vertices is a commuting graph of $ G $ for some people $ n $.

What is the minimum possible order of $ G $, such as $ UGT (G) = n $?

The constructs of the questions "Group Commutation Graphs" and "What is the minimum possible size of a graph $ n $ -universal?" We provide such a command group $ 2 ^ {(1 + o (1) 2 ^ { frac {n-1} {2}}} $.

However, a much smaller example (of order $ (2n (n-1))! $) is possible. C & # 39; $ S_ {2n (n-1)} $.

Indeed, let's suppose $ A = {a_ {ij} } _ {1 leq i, j leq n} cup {b_ {ij} } _ {1 leq i, j leq n} $. For a chart $ Gamma (V, E) $, or $ V = {1, …, n } $, we can take $ S_ Gamma subset Sym (A) $, defined by the following formula S $ {ji}) | i in V } $.

However, I do not know if there is still room for improvement …