Minimally Faithful Finite Group Actions

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I plan on highlighting some of the important open problems
left in finite group theory (namely, the Higman PORC Conjecture) and
then shifting tack to discuss faithful permutation representations
(long name, but it just means "injective homomorphisms G->S_n).  For a
finite group G, let d(G) be the smallest n such that there is an
injection G->S_n, and let alpha(G)=d(G)/#G, a rational number between
0 and 1.  I will discuss some of the key properties of alpha, with an
emphasis on p-groups, and display an intuitive algorithm to calculate
alpha for any group.  This leads naturally to the PORC-alpha
conjecture, a stronger conjecture than the Higman PORC conjecture and
yet somehow an easier one.  If I still have time, I will discuss some
other features of alpha, such as averages and limit points.