We know that it is usually much easier to work with highly symmetry objects, the objects that have many automorphisms like the sphere, Lie groups, complete graph,... But is there any advantage of asymmetric objects, the objects that have only trivial automorphisms and what is that advantage? For example, is there a category $C$ and property $P$ that true for all objects of $C$ that it's easier to prove assymetric objects have property $P$ in that category than for symmetric ones?
Motivation: Give a category $C$, we want to prove all objects of $C$ have property $P$. I have an interesting idea as follow:
For each objects $A$, let $G_A$ be a automorphisms group of $A$. Assume for each group morphism $H\rightarrow G_A$, we have a nice quotient $A/H$, and for some group $H$, it's easy to prove this statement "if $A/H$ has property $P$ then $A$ also has property $P$". For example, we can prove that statement for all cyclic groups, because for each non-trivial group $G$, there exists a non-trivial cyclic group as its subgroup. So we keep quotienting objects $A$, and if we are lucky, we will reach to an asymmetric object in $C$, and now we just need to prove that all asymmetric objects have property $P$.

We know that it is usually much easier to work with highly symmetry objects, the objects that have many automorphisms like the sphere, Lie groups, complete graph,... But is there any advantage of asymmetric objects, the objects that have only trivial automorphisms and what is that advantage? For example, is there a category $C$ and property $P$ that true for all objects of $C$ that it's easier to prove asymmetric objects have property $P$ in that category than for symmetric ones?
Motivation: Give a category $C$, we want to prove all objects of $C$ have property $P$. I have an interesting idea as follow:
For each objects $A$, let $G_A$ be a automorphisms group of $A$. Assume for each group morphism $H\rightarrow G_A$, we have a nice quotient $A/H$, and for some group $H$, it's easy to prove this statement "if $A/H$ has property $P$ then $A$ also has property $P$". For example, we can prove that statement for all cyclic groups, because for each non-trivial group $G$, there exists a non-trivial cyclic group as its subgroup. So we keep quotienting objects $A$, and if we are lucky, we will reach to an asymmetric object in $C$, and now we just need to prove that all asymmetric objects have property $P$.

We know that it is usually much easier to work with highly symmetry objects, the objects that have many automorphisms like the sphere, Lie groups, complete graph,... But is there any advantage of asymmetric objects, the objects that have only trivial automorphisms and what is that advantage? For example, is there a category $C$ and property $P$ that true for all objects of $C$ that it's easier to prove assymetric objects have property $P$ in that category than for symmetric ones?
Motivation: Give a category $C$, we want to prove all objects of $C$ have property $P$. I have an interesting idea as follow:
For each objects $A$, let $G_A$ be a automorphisms group of $A$. Assume for each group morphism $H\rightarrow G_A$, we have a nice quotient $A/H$, and for some group $H$, it's easy to prove this statement "if $A/H$ has property $P$ then $A$ also has property $P$". For example, we can prove that statement for all cyclic groups, because for each non-trivial group $G$, there exists a non-trivial cyclic subgroup. So we keep quotienting objects $A$, and if we are lucky, we will reach to an asymmetric object in $C$, and now we just need to prove that all asymmetric objects have property $P$.

We know that it is usually much easier to work with highly symmetry objects, the objects that have many automorphisms like the sphere, Lie groups, complete graph,... But is there any advantage of asymmetric objects, the objects that have only trivial automorphisms and what is that advantage? For example, is there a category $C$ and property $P$ that true for all objects of $C$ that it's easier to prove assymetric objects have property $P$ in that category than for symmetric ones?
Motivation: Give a category $C$, we want to prove all objects of $C$ have property $P$. I have an interesting idea as follow:
For each objects $A$, let $G_A$ be a automorphisms group of $A$. Assume for each group morphism $H\rightarrow G_A$, we have a nice quotient $A/H$, and for some group $H$, it's easy to prove this statement "if $A/H$ has property $P$ then $A$ also has property $P$". For example, we can prove that statement for all cyclic groups, because for each non-trivial group $G$, there exists a non-trivial cyclic group as its subgroup. So we keep quotienting objects $A$, and if we are lucky, we will reach to an asymmetric object in $C$, and now we just need to prove that all asymmetric objects have property $P$.