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Generating pairs of projective special linear groups that fail to lift

Boschheidgen, Jan ; Klopsch, Benjamin and Thillaisundaram, Anitha LU (2020) In Mathematische Nachrichten 293(7). p.1251-1258
Abstract
The following problem was originally posed by B. H. Neumann and H. Neumann.Suppose that a group can be generated by elements and that is a homomor-phic image of . Does there exist, for every generating -tuple (ℎ1, … , ℎ ) of ,a homomorphism ∶ → and a generating -tuple (1, … , ) of such that( 1 , … , ) = (ℎ1, … , ℎ )?M. J. Dunwoody gave a negative answer to this question, by means of a carefullyengineered construction of an explicit pair of soluble groups. Via a new approach weproduce, for = 2, infinitely many pairs of groups (, ) that are negative examplesto Neumanns’ problem. These new examples are easily described: is a free productof two suitable finite cyclic groups, such as 2 ∗ 3, and is a suitable finite projec-tive... (More)
The following problem was originally posed by B. H. Neumann and H. Neumann.Suppose that a group can be generated by elements and that is a homomor-phic image of . Does there exist, for every generating -tuple (ℎ1, … , ℎ ) of ,a homomorphism ∶ → and a generating -tuple (1, … , ) of such that( 1 , … , ) = (ℎ1, … , ℎ )?M. J. Dunwoody gave a negative answer to this question, by means of a carefullyengineered construction of an explicit pair of soluble groups. Via a new approach weproduce, for = 2, infinitely many pairs of groups (, ) that are negative examplesto Neumanns’ problem. These new examples are easily described: is a free productof two suitable finite cyclic groups, such as 2 ∗ 3, and is a suitable finite projec-tive special linear group, such as PSL(2, ) for a prime ≥ 5. A small modificationyields the first negative examples (, ) with infinite. (Less)
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author
; and
publishing date
type
Contribution to journal
publication status
published
subject
in
Mathematische Nachrichten
volume
293
issue
7
pages
1251 - 1258
publisher
John Wiley & Sons Inc.
external identifiers
  • scopus:85084801337
ISSN
1522-2616
DOI
10.1002/mana.201900354
language
English
LU publication?
no
id
b89c2704-ef28-414d-8507-2ecab77c8851
date added to LUP
2024-06-07 14:25:05
date last changed
2024-08-07 10:07:32
@article{b89c2704-ef28-414d-8507-2ecab77c8851,
  abstract     = {{The following problem was originally posed by B. H. Neumann and H. Neumann.Suppose that a group  can be generated by  elements and that  is a homomor-phic image of . Does there exist, for every generating -tuple (ℎ1, … , ℎ ) of ,a homomorphism  ∶  →  and a generating -tuple (1, … ,  ) of  such that( 1 , … ,  ) = (ℎ1, … , ℎ )?M. J. Dunwoody gave a negative answer to this question, by means of a carefullyengineered construction of an explicit pair of soluble groups. Via a new approach weproduce, for  = 2, infinitely many pairs of groups (, ) that are negative examplesto Neumanns’ problem. These new examples are easily described:  is a free productof two suitable finite cyclic groups, such as 2 ∗ 3, and  is a suitable finite projec-tive special linear group, such as PSL(2, ) for a prime  ≥ 5. A small modificationyields the first negative examples (, ) with  infinite.}},
  author       = {{Boschheidgen, Jan and Klopsch, Benjamin and Thillaisundaram, Anitha}},
  issn         = {{1522-2616}},
  language     = {{eng}},
  number       = {{7}},
  pages        = {{1251--1258}},
  publisher    = {{John Wiley & Sons Inc.}},
  series       = {{Mathematische Nachrichten}},
  title        = {{Generating pairs of projective special linear groups that fail to lift}},
  url          = {{http://dx.doi.org/10.1002/mana.201900354}},
  doi          = {{10.1002/mana.201900354}},
  volume       = {{293}},
  year         = {{2020}},
}