A phrase normally is a part of a sentence, It is not always true, of course, Many people would view "Sh*t!" as a phrase. But according to Wikipedia, phrase is usually required to include all the dependents of the units that it contains. Some expressions that may be called phrases in everyday language are not phrases in the technical sense. For example, in the sentence I can't put up with Alex, the words put up with may be referred to in common language as a phrase but they do not form a complete phrase, since they do not include Alex, which is the complement of the preposition with.

It is not a phrase.

All fundamental groups of closed 3-manifolds are linear, so they have word problem in logspace. Sol has exponential Dehn function, Nil has Dehn function $\sim n^3$.

Transitivity is not the same as ergodicity. I am not sure that if the quotient by the center is not taken, the action is ergodic.

Don't they usually also take a quotient by a compact (central) subgroup?

So what is the answer?

$Aut(S_4)=Inn(S_4)=S_4$. Two permutations are conjugate iff they have the same cycle structure.

Is the 3-manifold closed?

It would be interesting to know examples of non-real world graphs.

Every finite group is the automorphism group of a graph.

It is now less understandable. There is no such thing as real world graph. The other properties hold in a complete graph.

My intention was to show that, as formulated, the question does not make sense.

If the graph is complete then any random subgraph is complete. If the graph is a cycle (path) and $n$ is small enough then the subgraph will be empty with positive probability.

... I mean $\sigma_2(n)<e^{2\gamma} n^2(\log\log n)^2$ could be equivalent to RH.

...or, perhaps the simple $\sigma_1\le \sigma_2$ is enough.

No but perhaps you can get $\ge$ using Cauchy's inequality.

Can you use the fact that $\sigma_2(n)\le (\sigma_1(n))^2$?

here is a reference arxiv.org/pdf/1904.08297.pdf

What if $f(n)=n$?