A note about solvable and non-solvable finite groups of the same order type

2024/08/14 by Müller, Peter
#20D60 (Primary) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2408.07732

Abstract

Two finite groups are said to have the same order type if for each positive integer n both groups have the same number of elements of order n. In 1987 John G. Thompson asked if in this case the solvability of one group implies the solvability of the other group. In 2024 Pawel Piwek gave a negative example. He constructed two groups of order 2³⁶⁵⋅3¹⁰⁵⋅7¹⁰⁴≈7.3⋅10²⁴⁷ of the same order type, where only one is solvable. In this note we produce a much smaller example of order 2¹³⋅3⁴⋅7³=227598336.

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