# Free Action of Finite Groups on Spaces of Cohomology Type (0, b)

## TL;DR

This paper investigates conditions under which finite groups can act freely on spaces with specific cohomology types, revealing restrictions based on group structure and space dimension.

## Contribution

It establishes new restrictions on finite group actions on spaces of cohomology type (0, b), especially for types (0, 1) and (0, 0), depending on the dimension and group properties.

## Key findings

- Groups containing Zp + Zp + Zp cannot act freely on certain spaces.
- For spaces of type (0, 0), p-subgroups are cyclic or generalized quaternion.
- Z2 is the only group acting freely on spaces of type (0, 0) when n is even.

## Abstract

Let G be a finite group acting freely on a finitistic space X having cohomology type (0, b) (for example, S^n x S^{2n} is a space of type (0, 1) and the one-point union S^n V S^{2n} V S^{3n} is a space of type (0, 0)). It is known that a finite group G which contains Zp + Zp + Zp, p a prime, can not act freely on S^n x S^{2n}. In this paper, we show that if a finite group G acts freely on a space of type (0, 1), where n is odd, then G can not contain Zp + Zp, p an odd prime. For spaces of cohomology type (0, 0), we show that every p-subgroup of G is either cyclic or a generalized quaternion group. Moreover, for n even, it is shown that Z2 is the only group which can act freely on X.