Details
| Original language | English |
|---|---|
| Pages (from-to) | 9-18 |
| Number of pages | 10 |
| Journal | Archiv der Mathematik |
| Volume | 117 |
| Issue number | 1 |
| Early online date | 3 Mar 2021 |
| Publication status | Published - Jul 2021 |
Abstract
Motivated by recent results on the minimal base of a permutation group, we introduce a new local invariant attached to arbitrary finite groups. More precisely, a subset Δ of a finite group G is called a p-base (where p is a prime) if ⟨ Δ ⟩ is a p-group and C G(Δ) is p-nilpotent. Building on results of Halasi–Maróti, we prove that p-solvable groups possess p-bases of size 3 for every prime p. For other prominent groups, we exhibit p-bases of size 2. In fact, we conjecture the existence of p-bases of size 2 for every finite group. Finally, the notion of p-bases is generalized to blocks and fusion systems.
Keywords
- Base, Fusion, p-nilpotent centralizer
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: Archiv der Mathematik, Vol. 117, No. 1, 07.2021, p. 9-18.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Generalized bases of finite groups
AU - Sambale, Benjamin
N1 - Funding Information: The author is supported by the German Research Foundation (SA 2864/1-2 and SA 2864/3-1).
PY - 2021/7
Y1 - 2021/7
N2 - Motivated by recent results on the minimal base of a permutation group, we introduce a new local invariant attached to arbitrary finite groups. More precisely, a subset Δ of a finite group G is called a p-base (where p is a prime) if ⟨ Δ ⟩ is a p-group and C G(Δ) is p-nilpotent. Building on results of Halasi–Maróti, we prove that p-solvable groups possess p-bases of size 3 for every prime p. For other prominent groups, we exhibit p-bases of size 2. In fact, we conjecture the existence of p-bases of size 2 for every finite group. Finally, the notion of p-bases is generalized to blocks and fusion systems.
AB - Motivated by recent results on the minimal base of a permutation group, we introduce a new local invariant attached to arbitrary finite groups. More precisely, a subset Δ of a finite group G is called a p-base (where p is a prime) if ⟨ Δ ⟩ is a p-group and C G(Δ) is p-nilpotent. Building on results of Halasi–Maróti, we prove that p-solvable groups possess p-bases of size 3 for every prime p. For other prominent groups, we exhibit p-bases of size 2. In fact, we conjecture the existence of p-bases of size 2 for every finite group. Finally, the notion of p-bases is generalized to blocks and fusion systems.
KW - Base
KW - Fusion
KW - p-nilpotent centralizer
UR - http://www.scopus.com/inward/record.url?scp=85102053186&partnerID=8YFLogxK
U2 - 10.1007/s00013-021-01589-x
DO - 10.1007/s00013-021-01589-x
M3 - Article
AN - SCOPUS:85102053186
VL - 117
SP - 9
EP - 18
JO - Archiv der Mathematik
JF - Archiv der Mathematik
SN - 0003-889X
IS - 1
ER -