Details

Autor(en) / Beteiligte

• Wang, Yu Lei
• Liu, He Guo

Titel

Automorphisms of a Class of Finite p-groups with a Cyclic Derived Subgroup

Ist Teil von

• Acta mathematica Sinica. English series, 2021-06, Vol.37 (6), p.926-940

Ort / Verlag

Beijing: Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society

Erscheinungsjahr

2021

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• Let p be an odd prime, and let k be a nonzero nature number. Suppose that nonabelian group G is a central extension as follows where G ′ ≅ ℤ p k , and ζG/G ′ is a direct factor of G / G ′.Then G is a central product of an extraspecial p k -group E and ζG . Let ∣ E ∣ = p (2 n +1) k and ∣ ζG ∣ = p ( m +1) k . Suppose that the exponents of E and ζG are p k+l and p k+r , respectively, where 0 ≤ l, r ≤ k . Let Aut G ′ G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G ′, let Aut G/ζG,ζG G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the center ζG , and let Aut G/ζG,ζG/G ′ G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on ζG/G ′. Then (i) The group extension 1 → Aut G ′ G → Aut G → Aut G ′ → 1 is split. (ii) Aut G ′ G /Aut G/ζG,ζG G ≅ G 1 × G 2 , where , H is an extraspecial p k -group of order p (2 n −1) k and . In particular, if and only if l = k and r = 0; if and only if l ≤ r ; if and only if r = k ; if and only if r = 0. (iii) Aut G ′ G/ Aut G/ζG,ζG/G ′ G ≅ G 1 × G 3 , where G 1 is defined in (ii); . In particular, if and only if r = k ; if and only if r = 0. (iv) . If m = 0, then ; If m > 0, then , and .