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

Caranti, A.; Tsang, Cindy

doi:10.1016/j.jalgebra.2022.11.013

Cited by 2 publications

(8 citation statements)

References 14 publications

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“…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…”

Section: It Was Shown In [Ct23 Section 4] Thatmentioning

confidence: 88%

“…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.…”

Section: Notationmentioning

confidence: 98%

“…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.…”

Section: Notationmentioning

confidence: 99%

“…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.…”

Section: Multiple Holomorph Via Bilinear Formsmentioning

confidence: 99%

“…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 :…”

Section: It Was Shown In [Ct23 Section 4] Thatmentioning

confidence: 99%

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

Caranti¹,

Tsang²

2023

Preprint

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We consider the quotient group T (G) of the multiple holomorph by the holomorph of a finite p-group G of class two for an odd prime p. By work of the first-named author, we know that T (G) contains a cyclic subgroup of order p r−1 (p − 1), where p r is the exponent of the quotient of G by its center. In this paper, we shall exhibit examples of G (with r = 1) such that T (G) has order exactly p − 1, which is as small as possible.

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“…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…”

Section: It Was Shown In [Ct23 Section 4] Thatmentioning

confidence: 88%

Section: Notationmentioning

confidence: 98%

Section: Notationmentioning

confidence: 99%

“…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.…”

Section: Multiple Holomorph Via Bilinear Formsmentioning

confidence: 99%

Section: It Was Shown In [Ct23 Section 4] Thatmentioning

confidence: 99%

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

Caranti¹,

Tsang²

2023

Preprint

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

Caranti,

Tsang

2023

Journal of Group Theory

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We consider the quotient group T ⁢ ( G ) T(G) of the multiple holomorph by the holomorph of a finite 𝑝-group 𝐺 of class two for an odd prime 𝑝. By work of the first-named author, we know that T ⁢ ( G ) T(G) contains a cyclic subgroup of order p r − 1 ⁢ ( p − 1 ) p^{r-1}(p-1) , where p r p^{r} is the exponent of the quotient of 𝐺 by its center. In this paper, we shall exhibit examples of 𝐺 (with r = 1 r=1 ) such that T ⁢ ( G ) T(G) has order exactly p − 1 p-1 , which is as small as possible.

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

Cited by 2 publications

References 14 publications

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

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

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

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