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Let $G$ be an infinite group such that for every $g$$g\neq 1$ in $G$, there is some $b$ such that $C(b)$ has finite index in $G$ and $g$ is not in $C(b)$. Is something known about those groups? Are they FC groups, or related to FC groups?
Let $G$ be an infinite group such that for every $g$ in $G$, there is some $b$ such that $C(b)$ has finite index in $G$ and $g$ is not in $C(b)$. Is something known about those groups? Are they FC groups, or related to FC groups?
Let $G$ be an infinite group such that for every $g\neq 1$ in $G$, there is some $b$ such that $C(b)$ has finite index in $G$ and $g$ is not in $C(b)$. Is something known about those groups? Are they FC groups, or related to FC groups?
Groups with trivial centralizer-connected component
Let $G$ be an infinite group such that for every $g$ in $G$, there is some $b$ such that $C(b)$ has finite index in $G$ and $g$ is not in $C(b)$. Is something known about those groups? Are they FC groups, or related to FC groups?
