Almost all loops have a trivial automorphism group; almost all groups have a non-trivial automorphism group. What goes on in between?

2012-07-14T14:41:50Z

NB: For this question, everything is finite.

Recently I've been fascinated by the following two observations:

Almost all loops have a trivial automorphism group (McKay & Wanless, 2005, in the context of Latin squares).
Almost all groups have a non-trivial automorphism group (in fact, all groups of order $n \geq 3$ admit a non-trivial automorphism).

However, the only difference between loops and groups is "cancellation" vs. "associativity". (Actually, in a loop, an element's left inverse might not equal its right inverse, but the equality of left and right inverses in a group does not need to be included as a group axiom.)

Question: How can we strengthen the loop axioms so as to preserve the property that "almost all X have a trivial automorphism group"?

Question: How can we weaken the group axioms so as to preserve the property that "almost all Y have a non-trivial automorphism group"?

Question: Is the a set of axioms "between" the loop axioms and the group axioms for which "almost all Z have a trivial automorphism group" and "almost all Z have a non-trivial automorphism group" are both false?

Answer by Gerhard Paseman for Almost all loops have a trivial automorphism group; almost all groups have a non-trivial automorphism group. What goes on in between?

2012-07-14T17:31:47Z

I suggest a general algebraic approach for this problem, since to me the major thing that is changing is the idea of 'many". Although you haven't said so, you seem to be asking about a demarcation in the lattice of varieties of algebra with one binary operation where one side has algebras with more than one automorphism versus those that have only one. The latter are called rigid, and since no nontrivial variety has only rigid algebras (think of powers), you will need to have a good technical definition of 'many'. Perhaps pseudovarieties are the classes of interest.

I recommend looking at studies of rigid algebras. If you are interested in equational formulations which promote rigidity, you could do worse than looking at versions of primal algebras, which are very rigid. The varieties they generate are called arithmetical, and therein might lie part of the answer you seek.

Gerhard "Ask Me About System Design" Paseman, 2012.07.14

Almost all loops have a trivial automorphism group; almost all groups have a non-trivial automorphism group. What goes on in between? - MathOverflow

Almost all loops have a trivial automorphism group; almost all groups have a non-trivial automorphism group. What goes on in between?

Answer by Gerhard Paseman for Almost all loops have a trivial automorphism group; almost all groups have a non-trivial automorphism group. What goes on in between?