## Abstract

Given a finite group G and two unitary G-representations V and W, possible restrictions on topological degrees of equivariant maps between representation spheres S(V) and S(W) are usually expressed in a form of congruences modulo the greatest common divisor of lengths of orbits in S(V) (denoted by α(V)). Effective applications of these congruences is limited by answers to the following questions: (i) under which conditions, is α(V)>1? and (ii) does there exist an equivariant map with the degree easy to calculate? In the present paper, we address both questions. We show that α(V)>1 for each irreducible non-trivial C[G]-module if and only if G is solvable. This provides a new solvability criterion for finite groups. For non-solvable groups, we use 2-transitive actions to construct complex representations with non-trivial α-characteristic. Regarding the second question, we suggest a class of Norton algebras without 2-nilpotents giving rise to equivariant quadratic maps, which admit an explicit formula for the degree.

Original language | English |
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Pages (from-to) | 45-77 |

Number of pages | 33 |

Journal | Journal of Algebra |

Volume | 549 |

DOIs | |

State | Published - 1 May 2020 |

## Keywords

- Doubly transitive groups
- Equivariant map
- Norton algebras
- Ordinary representations of finite groups
- Solvable groups
- Topological degree

## ASJC Scopus subject areas

- Algebra and Number Theory