### Abstract

Let G be a finite group. A gap G-module V is a finite dimensional real G-representation space satisfying the following two conditions: (1) The following strong gap condition holds: dim V^{P}>2\dim V^{H} for all P < H≤ G such that P is of prime power order, which is a sufficient condition to define a G-surgery obstruction group and a G-surgery obstruction. (2) V has only one H-fixed point 0 for all large subgroups H, namely H ∈ ℒmathscr(L)(G). A finite group G not of prime power order is called a gap group if there exists a gap G-module. We discuss the question when the direct product K× L is a gap group for two finite groups K and L. According to [(5)], if K and K× C_{2} are gap groups, so is K× L. In this paper, we prove that if K is a gap group, so is K× C_{2}. Using [(5)], this allows us to show that if a finite group G has a quotient group which is a gap group, then G itself is a gap group. Also, we prove the converse: if K is not a gap group, then K× D_{2n} is not a gap group. To show this we define a condition, called NGC, which is equivalent to the non-existence of gap modules.

Original language | English |
---|---|

Pages (from-to) | 975-990 |

Number of pages | 16 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 53 |

Issue number | 4 |

DOIs | |

Publication status | Published - Jan 1 2001 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

**Gap modules for direct product groups.** / Sumi, Toshio.

Research output: Contribution to journal › Article

*Journal of the Mathematical Society of Japan*, vol. 53, no. 4, pp. 975-990. https://doi.org/10.2969/jmsj/05340975

}

TY - JOUR

T1 - Gap modules for direct product groups

AU - Sumi, Toshio

PY - 2001/1/1

Y1 - 2001/1/1

N2 - Let G be a finite group. A gap G-module V is a finite dimensional real G-representation space satisfying the following two conditions: (1) The following strong gap condition holds: dim VP>2\dim VH for all P < H≤ G such that P is of prime power order, which is a sufficient condition to define a G-surgery obstruction group and a G-surgery obstruction. (2) V has only one H-fixed point 0 for all large subgroups H, namely H ∈ ℒmathscr(L)(G). A finite group G not of prime power order is called a gap group if there exists a gap G-module. We discuss the question when the direct product K× L is a gap group for two finite groups K and L. According to [(5)], if K and K× C2 are gap groups, so is K× L. In this paper, we prove that if K is a gap group, so is K× C2. Using [(5)], this allows us to show that if a finite group G has a quotient group which is a gap group, then G itself is a gap group. Also, we prove the converse: if K is not a gap group, then K× D2n is not a gap group. To show this we define a condition, called NGC, which is equivalent to the non-existence of gap modules.

AB - Let G be a finite group. A gap G-module V is a finite dimensional real G-representation space satisfying the following two conditions: (1) The following strong gap condition holds: dim VP>2\dim VH for all P < H≤ G such that P is of prime power order, which is a sufficient condition to define a G-surgery obstruction group and a G-surgery obstruction. (2) V has only one H-fixed point 0 for all large subgroups H, namely H ∈ ℒmathscr(L)(G). A finite group G not of prime power order is called a gap group if there exists a gap G-module. We discuss the question when the direct product K× L is a gap group for two finite groups K and L. According to [(5)], if K and K× C2 are gap groups, so is K× L. In this paper, we prove that if K is a gap group, so is K× C2. Using [(5)], this allows us to show that if a finite group G has a quotient group which is a gap group, then G itself is a gap group. Also, we prove the converse: if K is not a gap group, then K× D2n is not a gap group. To show this we define a condition, called NGC, which is equivalent to the non-existence of gap modules.

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U2 - 10.2969/jmsj/05340975

DO - 10.2969/jmsj/05340975

M3 - Article

VL - 53

SP - 975

EP - 990

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

IS - 4

ER -