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Recall that a group $G$ is residualy finite if for every non-zero element $g\in G$ there exists a homomorphism $\sigma:G\rightarrow H$ such that $H$ is finite and $\sigma(g)\neq 0$. Mal'cev's theorem …

Representation groups are a nice example. If $G$ is a finite group of order $n$ and if $m$ is the order of $H^2(G,\mathbb{C}^*)$, then a representation group is a group written as a central extension …

Note that it is not hard to construct examples where $Br(G)$ is not equal to $M(G)$. To see this, let $H_1=\mathbb{Z}/p\times\mathbb{Z}/p$, and let $\alpha_1\in H^2(H_1,\mathbb{C}^*)=\mathbb{Z}/p$ be …

Let $G$ be a group, and let $M(G)=H^2(G,\mathbb{C}^*)$ be the Schur multiplier of $G$. There is a group $Br(G)$ of complex projective representations of $G$ modulo those that can be lifted to linear r …