A BOUND ON THE ORDER OF THE OUTER AUTOMORPHISM GROUP OF A FINITE SIMPLE GROUP OF GIVEN ORDER

STEFAN KOHL

Abstract. We prove that the order of the outer automorphism group of a non-abelian finite simple group of order n is bounded by_{2} log n, and show that this bound is sharp in the sense that it cannot be improved by more than a small constant factor. The proof uses the classification of finite simple groups.

Theorem 1. Let G be a non-abelian finite simple group. Then we have

|Out(G)| < _{2} log |G|.

Proof. We use the classification of finite simple groups (see e.g. [1], in particular p. xvi, Table 5, 6). In the following, let q = p^{f} denote a prime power and n a positive integer.

(1) Let G be an alternating group. Then we have |Out(G)| ≤ 4 < _{2} log 60 ≤ 2 log |G|.

(2) Let G be either one of the 26 sporadic simple groups or the Tits group. Then we have |Out(G)| ≤ 2 < _{2} log 7920 ≤ _{2} log |G|.

(3) Assume G ≅ A_{1}(q) ≅ PSL(2, q). Then we have

|Out(G)| ≤ 2f ≤ 2 ·_{2} log q = _{2} log(q^{2})

<^{q⩾4} _{2} log (\frac{1}{2} q(q^{2} − 1)) ≤ _{2} log |G|.

(4) Assume G ≅ A_{2}(q) ≅ PSL(3, q). Then we have

|Out(G)| ≤ 6f ≤ 6 ·_{2} log q = _{2} log(q^{6})

<_{2} log (\frac{1}{(3, q − 1)} q^{3}(q^{2} − 1)(q^{3} − 1)) = _{2} log |G|.

(5) Assume G ≅ ^{2}A_{2}(q) ≅ PSU(3, q). Then we have

|Out(G)| ≤ 6f ≤ 6 ·_{2} log q = _{2} log(q^{6})

<_{2} log (\frac{1}{(3, q + 1)} q^{3}(q^{2} − 1)(q^{3} + 1)) = _{2} log |G|.

(6) Assume G ≅ A_{3}(q) ≅ PSL(4, q) or G ≅ ^{2}A_{3}(q) ≅ PSU(4, q). Then we have

|Out(G)| ≤ 8f ≤ 8 ·_{2} log q = _{2} log(q^{8})

<_{2} log (\frac{1}{4} q^{6}(q^{2} − 1)(q^{3} − 1)(q^{4} − 1)) ≤ _{2} log |G|.

2000 Mathematics Subject Classification. Primary 20E32, 20D06, 20D45.

(7) Assume G ≅ A_{n}(q) ≅ PSL(n + 1, q) or G ≅ ^{2}A_{n}(q) ≅ PSU(n + 1, q), where n ≥ 4. Then we have

|Out(G)| ≤ 2(n + 1)f ≤ 2(n + 1) ·_{2} log q
≤ \frac{n(n + 1)}{2} ·_{2} log q = _{2} log (q^{\frac{n(n+1)}{2}} )
≤_{2} log (\frac{1}{q + 1} q^{\frac{n(n+1)}{2}} (q^{2} − 1)) < _{2} log |G|.

(8) Assume G ≅ B_{n}(q) ≅ O(2n + 1, q). Then we have

|Out(G)| ≤ 2f ≤ 2 ·_{2} log q
< n^{2} ·_{2} log q = _{2} log (q^{n^{2}}) < _{2} log |G|.

(9) Assume G ≅ ^{2}B_{2}(q) ≅ Sz(q). Then we have |Out(G)| = _{2} log q < _{2} log |G|.

(10) Assume G ≅ C_{n}(q) ≅ PSp(2n, q). Then we have

|Out(G)| ≤ 2f ≤ 2 ·_{2} log q
< n^{2} ·_{2} log q = _{2} log (q^{n^{2}}) < _{2} log |G|.

(11) Assume G ≅ D_{4}(q) ≅ O^{+}(8, q). Then we have

|Out(G)| ≤ 24f ≤ 24 ·_{2} log q = _{2} log(q^{24})
<_{2} log (\frac{1}{(4, q^{4} − 1)} q^{12}(q^{4} − 1)(q^{2} − 1)(q^{4} − 1)(q^{6} − 1))
=_{2} log |G|.

(12) Assume G ≅ D_{n}(q) ≅ O^{+}(2n, q) where n ≥ 5, or G ≅ ^{2}D_{n}(q) ≅ O^{−}(2n, q) where n ≥ 4. Then we have

|Out(G)| ≤ 8f ≤ 8 ·_{2} log q
<^{n⩾4} n^{2} ·_{2} log q = _{2} log (q^{n^{2}}) < _{2} log |G|.

(13) Assume G ∈ {^{3}D_{4}(q),G_{2}(q),^{2}G_{2}(q),F_{4}(q),^{2}F_{4}(q),E_{6}(q),^{2}E_{6}(q),E_{7}(q),E_{8}(q)}.

Then we have |Out(G)| ≤ 6f ≤ 6 ·_{2} log q = _{2} log(q^{6}) < _{2} log |G|. □

Remark 2. The bound in Theorem 1 cannot be improved significantly. If f is even and q = 2^{f}, we have

\frac{|Out(A_{2}(q))|}{_{2} log |A_{2}(q)|} = \frac{6f}{_{2} log (\frac{1}{3} q^{3}(q^{2} − 1)(q^{3} − 1))} ⟶_{f→∞} \frac{3}{4}.

Hence we even cannot replace our bound by ln |G| – even not if we allow a finite number of exceptions.

References

1. John H. Conway, Robert T. Curtis, Simon P. Norton, Richard A. Parker, and Robert A. Wilson, Atlas of finite groups, Oxford University Press, 1985.

Institut für Geometrie und Topologie, Pfaffenwaldring 57, Universität Stuttgart 70550 Stuttgart, Germany

E-mail address: kohl@mathematik.uni-stuttgart.de