Rational groups

Question

A finite group $G$ is called rational if and only if $N_G(\langle x\rangle)/C_G(x)\cong Aut(\langle x\rangle)$ for all $x\in G$. The word ``rational" is because there is an equivalent definition in group representation theoretic terms: A finite group $G$ is rational if and only if for any complex irreducible character $\chi$ of $G$, $\chi(g)$ is rational for all $g\in G$.

Is it true that a finite group $G$ is rational if $N_G(\langle x\rangle)/C_G(x)\cong Aut(\langle x\rangle)$ for any $p$-element $x$ of $G$ and for all primes $p$ dividing $|G|$.

Answer 1

No, this is not true, and the smallest counterexample is the dihedral group of order 24. The condition is trivially satisfied for all $x\in G$ of order $2$, and it also holds for rotations of order $3$ and $4$, since these are conjugate to their inverses. But rotations of order $12$ have $N/C$ of size 2, rather than 4.

If you want, the following Magma script (that you can stick into the Magma calculator) computes all counterexamples of order up to $100$.

for n in [2..100], G in SmallGroups(n) do
  HasLargeAut:=func<g|forall{i: i in [1..Order(g)] |
     (GCD(i,Order(g)) ne 1) or IsConjugate(G,g,g^i)}>;
  CC:={c[3]: c in ConjugacyClasses(G)};
  P:={c: c in CC | (Order(c) ne 1) and IsPrimePower(Order(c))};
  NP:=CC diff P;
  if forall{g: g in P | HasLargeAut(g)} and
    not forall{g: g in NP | HasLargeAut(g)} then
      IdentifyGroup(G);
  end if;
end for;