### Abstract

In this paper we construct nontrivial pairs of script G -related (i.e. Smith equivalent) real (G-modules for the group G = PΣL(2,27) and the small groups of order 864 and types 2666, 4666. This and a theorem of K. Pawalowski-R. Solomon together show that Laitinen's conjecture is affirmative for any finite nonsolvable gap group. That is, for a finite nonsolvable gap group G, there exists a nontrivial ℘(G)-matched pair consisting of G-related real G-modules if and only if the number of all real conjugacy classes of elements in G not of prime power order is greater than or equal to 2.

Original language | English |
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Pages (from-to) | 623-647 |

Number of pages | 25 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 62 |

Issue number | 2 |

DOIs | |

Publication status | Published - Apr 2010 |

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### Keywords

- Gap condition
- Laitinen's conjecture
- Representation
- Smith equivalence
- Tangent space

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Nontrivial ℘(G)-matched script G-related pairs for finite gap Oliver groups.** / Morimoto, Masaharu.

Research output: Contribution to journal › Article

*Journal of the Mathematical Society of Japan*, vol. 62, no. 2, pp. 623-647. https://doi.org/10.2969/jmsj/06220623

}

TY - JOUR

T1 - Nontrivial ℘(G)-matched script G-related pairs for finite gap Oliver groups

AU - Morimoto, Masaharu

PY - 2010/4

Y1 - 2010/4

N2 - In this paper we construct nontrivial pairs of script G -related (i.e. Smith equivalent) real (G-modules for the group G = PΣL(2,27) and the small groups of order 864 and types 2666, 4666. This and a theorem of K. Pawalowski-R. Solomon together show that Laitinen's conjecture is affirmative for any finite nonsolvable gap group. That is, for a finite nonsolvable gap group G, there exists a nontrivial ℘(G)-matched pair consisting of G-related real G-modules if and only if the number of all real conjugacy classes of elements in G not of prime power order is greater than or equal to 2.

AB - In this paper we construct nontrivial pairs of script G -related (i.e. Smith equivalent) real (G-modules for the group G = PΣL(2,27) and the small groups of order 864 and types 2666, 4666. This and a theorem of K. Pawalowski-R. Solomon together show that Laitinen's conjecture is affirmative for any finite nonsolvable gap group. That is, for a finite nonsolvable gap group G, there exists a nontrivial ℘(G)-matched pair consisting of G-related real G-modules if and only if the number of all real conjugacy classes of elements in G not of prime power order is greater than or equal to 2.

KW - Gap condition

KW - Laitinen's conjecture

KW - Representation

KW - Smith equivalence

KW - Tangent space

UR - http://www.scopus.com/inward/record.url?scp=77954339939&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77954339939&partnerID=8YFLogxK

U2 - 10.2969/jmsj/06220623

DO - 10.2969/jmsj/06220623

M3 - Article

VL - 62

SP - 623

EP - 647

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

IS - 2

ER -