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 $|G|$.

  • $\begingroup$ Isn't $Z/4$ a counterexample? $\endgroup$
    – M P
    Feb 3 '13 at 17:17
  • $\begingroup$ Btw, possibly I am confused: is your $g$ in the normalizer/centralizer formula an $x$? Also, by a $p$-element do you mean an element of order $p$ or of order a power of $p$? $\endgroup$
    – M P
    Feb 3 '13 at 17:28
  • $\begingroup$ @MP: If you mean by $Z/4$ the cyclic group of order $4$, it is not a counterexample. $\endgroup$ Feb 3 '13 at 17:29
  • 5
    $\begingroup$ The dihedral group of order 24 is a counterexample: elements of order 3 and 4 are conjugate to their inverses, but for an element of order 12 $N/C$ has size 2 rather than 4. $\endgroup$ Feb 3 '13 at 17:31
  • $\begingroup$ @MP: Thanks I changed $g$ to $x$ in the centralizers. By a $p$-element I mean an element whose order is a power of $p$. $\endgroup$ Feb 3 '13 at 17:32

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
  end if;
end for;

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