Betti number

in Group Cohomology for a finitely generated group G, the n-th Betti number $b_n(G)$ is defined as: $b_n(G)=di{m}_{\mathbb{Q}}{H}_n(G,\mathbb{Q})$ where $H_n(G,\mathbb{Q})$ is the n-th group homology of G with rational coefficients

Example

• $G={\mathbb{Z}}^n$ (free abelian group of rank n): $b_1(G)=n$
• $G=\mathbb{Z}/n\mathbb{Z}$ (finite cyclic group): $b_1(G)=0$
• $G={\mathbb{F}}_n$ (free group of rank n): $b_1(G)=n$