For ordinary groups, the question of trivial automorphism groups is settled: only the trivial group and Z/2Z have no nontrivial automorphisms. Every other group admits at least one non-identity self-map that preserves its structure. The result is classical, clean, almost obvious in retrospect — the more elements a group has, the more ways to rearrange them.
Skew braces break this intuition.
A skew brace carries two group operations — additive and multiplicative — connected by a compatibility condition. They arise naturally in the study of solutions to the Yang-Baxter equation. You might expect the additional algebraic structure to create more automorphisms, not fewer. Two operations mean more handles to grab; more structure should mean more symmetry.
Tsang constructs skew braces of order 2p³ (for any odd prime p) with no nontrivial automorphisms whatsoever. These are not tiny objects — they have 2p³ elements, carrying two intertwined group structures. Yet there is no non-identity map that preserves both operations simultaneously. The enrichment from one operation to two has locked the structure into a position so rigid that nothing can move.
The mechanism is the compatibility condition. Each operation alone would admit automorphisms. But the constraint linking them — the requirement that any map must preserve both simultaneously — eliminates every candidate. The two operations constrain each other into mutual immobility.
This is a structural principle that extends beyond algebra. Adding structure to a system does not generically increase its flexibility. When the added constraints interact, they can produce rigidity that neither constraint alone could achieve. Complexity is not always a source of richness. Sometimes it is a cage.
The enriched object is less symmetric than either of its parts.