ON THE ORDERS OF AUTOMORPHISM GROUPS OF FINITE GROUPS

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1 Submitted exclusively to the London Mathematical Society DOI: 0./S ON THE ORDERS OF AUTOMORPHISM GROUPS OF FINITE GROUPS JOHN N. BRAY and ROBERT A. WILSON Abstract In the Kourovka Notebook, Deaconescu asks if φ() for all finite groups G, where φ denotes the Euler totient function; and whether G is cyclic whenever = φ(). We answer both questions in the negative. Moreover we show that /φ() can be made arbitrarily small.. The question, and some answers Conventions. Throughout this paper, we shall only consider finite groups. The notation for group structures is based on that used in the Atlas []. The notation O (G), O (G) and Aut G is standard. Let φ denote the Euler totient function, so that φ(n) is the number of integers m with m n such that m and n are coprime, and φ(n) n = r i= p i p i, where p < p <... < p r are the prime factors of n. It is easy to see that for finite abelian groups G, we have φ(), with equality if and only if G is cyclic. In Problem 5.43 of the Kourovka Notebook [3], Deaconescu asks if the same is true for arbitrary finite groups G. More specifically: Let G be a finite group of order n. a) Is it true that φ(n) where φ is Euler s function? b) Is it true that G is cyclic if = φ(n)? In this note we show that the answer to both questions is no. Indeed, we shall prove: Main Theorem. For all ε > 0 there exists a group G such that < ε.φ(). In the course of this paper, it will transpire that there are infinitely many groups G satisfying < φ(), and infinitely many non-cyclic groups G which satisfy = φ(). 000 Mathematics Subject Classification 0E36 (primary), 0F8 (secondary).

2 john n. bray and robert a. wilson Part (a) Examining some quasisimple groups, we quickly found that the perfect groups G = (3 4 ) L 3 (4), M and (3 4) U 4 (3) satisfy < φ(), answering this part of the question in the negative. More precisely: group G primes dividing Aut G φ() (3 4 ) L 3 (4), 3, 5, 7 L 3 (4): 6 M, 3, 5, 7, M : 6 (3 4) U 4 (3), 3, 5, 7 U 4 (3):D Part (b) One non-cyclic group G satisfying = φ() is G = 3 5 M. Each of the five direct factors of G is characteristic, and we obtain Aut G = Aut Aut 3 Aut 5 Aut Aut M = 4 0 M, a group of order φ(). More generally, if.3.5. = 330 m, then the non-cyclic group G = C m M has automorphism group Aut G = (Aut C m ) M, and is thus a group of order φ(). The smallest non-cyclic group G we know of that satisfies = φ() is a group of order 56448, namely the group G = 4 :L 3 () 3 7, where the 4 :L 3 () is isomorphic to the centraliser of an involution in M 3. In the direct factor 4 :L 3 (), when we consider the normal 4 as an F -module for a complementary L 3 (), it is uniserial with a single non-zero proper submodule; this submodule has dimension. We have Aut G = 3 :L 3 () 6. We may take G to be the group g, g, g 3 where: g = (, 3)(, 4)(5, 5, 6, 6)(7, 9)(8, 0)(, 3,, 4), g = (, 3, 5)(, 4, 6)(7, 5, )(8, 6, ), g 3 = (7, 8, 9)(0,,, 3, 4, 5, 6). In fact, this group G is a special case of the groups we construct in the next section.. Proof of the Main Theorem Let P be a non-empty finite set of primes such that p 7 (mod 8) for all p P. We shall consider groups of the form G = C p P M p, where C is a cyclic subgroup of odd order and M p is a perfect group of shape f(p) :L (p). (We shall define the groups M p below.) Lemma. For all p P, the subgroups C and M p are characteristic in G. Thus Aut G = Aut C p P Aut M p.

3 on the orders of automorphism groups of finite groups 3 Proof. First, note that C = O (G), so is characteristic in G. Let N = F(G), the Fitting subgroup of G, so that N is also characteristic in G. Now G/N = p P L (p) is a direct product of non-isomorphic simple groups. For each p P, G/N has a unique normal subgroup N p /N such that N p /N = L (p). Thus N p /N is characteristic in G/N, and since N is characteristic in G, we get (for each p P ) that N p is a characteristic subgroup of G. In fact, N p = N, M p, and we have N p = C M p O (M q ). q P \{p} Since C and O (M q ) are abelian and M p is perfect, we obtain N p = M p, and thus M p is also a characteristic subgroup of G. Since C is cyclic, Aut C is an abelian group of order φ( C ). We now concentrate on the groups M p (where we may now fix p 7 (mod 8)). The order of Aut M p depends crucially on the structure of O (M p ) as an F L (p)-module. We aim to construct a uniserial module with composition factors of dimensions and (p ), so that M p has centre of order, in such a way that M p has trivial outer automorphism group. It will then follow that Aut M p = M p. In what follows, we shall use V V to denote a non-split extension of modules V by V where V is the submodule and V is the quotient. Lemma. Let H denote the simple group L (p), where p 7 (mod 8), and let V denote the permutation module over F of H on the p + cosets of the Borel subgroup. Then V is isomorphic to the trivial PIM (projective indecomposable module) for H, and has structure (U U ), where U and U are absolutely irreducible. Proof. First we need some notation for some elements of H. We use t to denote any element of order p; all elements of order p in H are conjugate to t or t. The other elements of H have order dividing (p ) or (p + ); all such non-identity elements can be notated by x, y or z where: x has order dividing (p ). y has order dividing 4 (p + ). z has order dividing (p + ), but does not have order dividing 4 (p + ). Four ordinary irreducible characters of L (p) are Element x y z t t χ 0 χ (p ) 0 β γ (p ) 0 γ β χ χ 3 p 0 0 where β and γ denote the irrationalities ( ± p); thus β and γ have minimal polynomial X +X+ 4 (p+). Since χ 0, χ and χ remain irreducible on restriction to the Borel subgroup of shape p:( p ) (which has odd order because p 3 (mod 4)), they remain irreducible on reduction modulo. Moreover, since p 7 (mod 8), X + X + 4 (p + ) reduces modulo to X + X, which has roots in F, so the

4 4 john n. bray and robert a. wilson corresponding representations can be written over F. For i {0,, } let ϕ i denote the -modular Brauer character which is the restriction of χ i to -regular classes, and let U i be a module affording the character ϕ i. Now let P() denote the PIM of the trivial representation in characteristic, and let Perm(p + ) denote the characteristic permutation module of L (p), of degree p +, on the cosets of the Borel subgroup p:( p p ). Since p:( ) has odd order, Perm(p + ) is projective, and thus, since it contains a trivial submodule, contains a copy of P(). Since the corresponding characteristic 0 permutation module of degree p + has character χ 0 + χ 3 and any element z is necessarily -singular, we see that Perm(p + ) has Brauer character ϕ 0 + ϕ + ϕ, and thus composition factors U 0, U 0, U, U. Now P() has a unique simple submodule, and unique simple quotient, and thus (since 4 L (p) ) has the form U where U is a non-zero module. Since U and U are conjugate under an outer automorphism of L (p) while U 0 = remains invariant, we obtain that P() = Perm(p + ) = (U U ). This structure is valid over any field of characteristic. Since modules with simple socle M embed in the PIM corresponding to M, there are unique F L (p)-modules of shapes U and U while there are no F L (p)- modules of shapes U or U. We now define M p = f(p) :L (p) to be the split extension of the F L (p)-module U by L (p); in particular f(p) = (p + ). It remains to prove: Lemma 3. With this definition, M p has trivial outer automorphism group. Proof. Since O (M p ) is a characteristic subgroup of M p, any automorphism of M p permutes the complements to O (M p ) in M p. Now let S denote a complementary L (p) in M p. We have ensured that the module U has zero -cohomology; thus M p has just one class of complements, so we may assume that our automorphism α of M p normalises S. Since the module U is not invariant under outer automorphisms of S, α S must be an inner automorphism of S, and adjusting by an inner automorphism of M p that is conjugation by an element of S, we may assume that α centralises S. So now α is an F S-module automorphism of U, and thus is a scalar, and therefore trivial. It follows that Aut M p = M p and therefore whence = φ( C ) C P φ() = φ( C ) C φ(). P Now if for all p P and odd prime divisors q of p(p + )(p ) we have that q C then it follows that φ( C )/ C = φ()/, and therefore φ() = φ( C ) C φ() = P P. In particular, this holds if C is the product of all the odd primes which divide p(p + )(p ) for some p P. To complete the proof of the Main Theorem,

5 on the orders of automorphism groups of finite groups 5 we invoke Dirichlet s Theorem that there are infinitely many primes p such that p 7 (mod 8) to conclude that P can be made arbitrarily large. 3. Further work We have now extended our constructions and have been able to show that the Main Theorem holds when G is restricted to being soluble, and also when G is restricted to being perfect. Moreover, there are infinitely many perfect groups G and infinitely many non-cyclic soluble groups G such that = φ(). These results are the subject of a forthcoming publication []. References. J. N. Bray and R. A. Wilson. On the orders of automorphism groups of finite groups. II. In preparation.. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson. An ATLAS of Finite Groups. Clarendon Press, Oxford (985; reprinted with corrections, 003). 3. E. I. Khukhro and V. D. Mazurov (Eds). Unsolved problems in group theory. The Kourovka Notebook, no. 5. Novosibirsk, 00. John N. Bray and Robert A. Wilson. School of Mathematics and Statistics, University of Birmingham, Edgbaston, Birmingham, B5 TT. jnb@maths.bham.ac.uk R.A.Wilson@bham.ac.uk