The automorphism group of a generalized extraspecial p-group

Abstract

In this paper, the automorphism group of a generalized extraspecial p-group G is determined, where p is a prime number. Assume that |G| = p ^2n+m and |ζG| = p ^m, where n ⩾ 1 and m ⩾ 2.

• (1) When p is odd, let Aut_G′ G = {α ∈ AutG | α acts trivially on G′}. Then Aut_G′ G ⊲ AutG and AutG/Aut_G′ G ≌ ℤ_p−1. Furthermore

• (i) If G is of exponent p ^m, then Aut_G′G/InnG ⊲ Sp(2n, p) × ℤ _p m−1.

• (ii) If G is of exponent p ^m+1, then Aut_G′ G/InnG ⊲ (K ⋊ Sp(2n − 2, p)) × ℤ _p m−1, where K is an extraspecial p-group of order p²ⁿ⁻¹.

In particular, Aut_G′ G/InnG ⊲ ℤ _p × ℤ _p m−1 when n = 1.

• (2) When p = 2, then

  • (i) If G is of exponent 2^m, then AutG ⊲ Sp(2n, 2) × ℤ₂ × ℤ₂ m−2.

    In particular, when n = 1, |AutG| = 3 · 2^m+2. None of the Sylow subgroups of AutG is normal, and each of the Sylow 2-subgroups of AutG is isomorphic to H ⋊ K, where H = ℤ₂ × ℤ₂ × ℤ₂ × ℤ₂ m−2, K = ℤ₂.

  • (ii) If G is of exponent 2^m+1, then AutG ⊲ (I ⋊ Sp(2n − 2, 2)) × ℤ₂ × ℤ₂ m−2, where I is an elementary abelian 2-group of order 2²ⁿ⁻¹.

In particular, when n = 1, |AutG| = 2^m+2 and AutG ⊲ H ⋊ K, where H = ℤ₂ × ℤ₂ × ℤ₂ m−1, K = ℤ₂.