Finite p-groups of class two with a large multiple holomorph

Finite $p$-groups of class two with a small multiple holomorph

“…Given any ϑ Hol(G) ∈ T (G) with 1 ϑ = 1, the exact same argument as in [CT23] shows that ϑ induces, via restriction, automorphisms…”

“…By [Car16], the condition Aut c (G) = 1 may be realized by certain π of full rank for any n ≥ 4. Using these special p-groups G and (1.1), we proved in [CT23] that T (G) can be made arbitrarily large.…”

“…In [CT23], we considered a family of p-groups G of class two which may be constructed from linear maps π : V → Λ 2 V . Here V is a finite dimensional vector space over F p and Λ 2 V denotes its exterior square.…”

“…Notice that then Aut c (G) ⊆ Aut z (G). In this case, by [CT23], the structure of T (G) may be studied using certain bilinear forms, as follows.…”

“…Notice that the coset representative ϑ is unique up to Aut(G). In the case that Aut c (G) = 1, both of the above are then independent of the choice of ϑ, so as in [CT23] one gets a well-defined homomorphism res :…”