### Abstract

For a finite group G, an L(G)-free gap G-module V is a finite dimensional real G-representation space satisfying the two conditions: (1) V ^{L} = 0 for any normal subgroup L of G with prime power index. (2) dim V ^{P} > 2 dim V ^{H} for any P < H ≤ G such that P is of prime power order. A finite group G not of prime power order is called a gap group if there is an L (G)-free gap G-module. We give a necessary and sufficient condition for that G is a gap group for a finite group G satisfying that G/[G, G] is not a 2-group, where [G, G] is the commutator subgroup of G.

Original language | English |
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Pages (from-to) | 91-106 |

Number of pages | 16 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 64 |

Issue number | 1 |

DOIs | |

Publication status | Published - Oct 5 2012 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

**The gap hypothesis for finite groups which have an abelian quotient group not of order a power of 2.** / Sumi, Toshio.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - The gap hypothesis for finite groups which have an abelian quotient group not of order a power of 2

AU - Sumi, Toshio

PY - 2012/10/5

Y1 - 2012/10/5

N2 - For a finite group G, an L(G)-free gap G-module V is a finite dimensional real G-representation space satisfying the two conditions: (1) V L = 0 for any normal subgroup L of G with prime power index. (2) dim V P > 2 dim V H for any P < H ≤ G such that P is of prime power order. A finite group G not of prime power order is called a gap group if there is an L (G)-free gap G-module. We give a necessary and sufficient condition for that G is a gap group for a finite group G satisfying that G/[G, G] is not a 2-group, where [G, G] is the commutator subgroup of G.

AB - For a finite group G, an L(G)-free gap G-module V is a finite dimensional real G-representation space satisfying the two conditions: (1) V L = 0 for any normal subgroup L of G with prime power index. (2) dim V P > 2 dim V H for any P < H ≤ G such that P is of prime power order. A finite group G not of prime power order is called a gap group if there is an L (G)-free gap G-module. We give a necessary and sufficient condition for that G is a gap group for a finite group G satisfying that G/[G, G] is not a 2-group, where [G, G] is the commutator subgroup of G.

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U2 - 10.2969/jmsj/06410091

DO - 10.2969/jmsj/06410091

M3 - Article

VL - 64

SP - 91

EP - 106

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

IS - 1

ER -