Dade's Conjecture for the Simple O'Nan Group

Abstract

Let G be the simple O’Nan group. It has order 29 · 34 · 5 · 73 · 11 · 19 · 31, it has Schur multiplier of order 3, and Aut G /G is cyclic of order 2. In this paper, we prove that Dade’s conjecture for G is true. Here we mean by saying the conjecture the inductive form of the conjecture (see [3]). According to [3], the inductive form is equivalent to the invariant projective form if the outer automorphism group has cyclic Sylow q-subgroups for each prime q. Thus, for the simple O’Nan group, it suffices to check the invariant projective form. Moreover, if a defect group of a block is cyclic, then the invariant projective form is proved to be true for this block in [4]. Thus, it suffices to treat the primes 2, 3, and 7. In this paper, A B and A ·B denote a split and a nonsplit extension of A by B, respectively. We use n and Dn to denote a cyclic and a dihedral group of order n, respectively. Moreover, for a prime p, an elementary abelian group of order p is denoted by p. For an odd prime p, we denote by p1+2n + an extraspecial group of order p1+2n and exponent p, and by 21+2n − a minus-type extraspecial group of order 21+2n. Finally, Sn and An denote the symmetric group and the alternating group of degree n, respectively.