Let \$w\$ be a group-word. For a group \$G\$, let \$G_w\$ denote the set of all \$w\$-values in \$G\$ and \$w(G)\$ the verbal subgroup of \$G\$ corresponding to \$w\$. The word \$w\$ is semiconcise if the subgroup \$[w(G),G]\$ is finite whenever \$G_w\$ is finite. The group \$G\$ is an \$FC(w)\$-group if the set of conjugates \$x^{G_w}\$ is finite for all \$xin G\$. We prove that if \$w\$ is a semiconcise word and \$G\$ is an \$FC(w)\$-group, then the subgroup \$[w(G),G]\$ is \$FC\$-embedded in \$G\$, that is, the intersection \$C_G(x)cap [w(G),G]\$ has finite index in \$[w(G),G]\$ for all \$xin G\$. A similar result holds for \$BFC(w)\$-groups, that are groups in which the sets \$x^{G_w}\$ are boundedly finite. We also show that this is no longer true if \$w\$ is not semiconcise.

### On semiconcise words

#### Abstract

Let \$w\$ be a group-word. For a group \$G\$, let \$G_w\$ denote the set of all \$w\$-values in \$G\$ and \$w(G)\$ the verbal subgroup of \$G\$ corresponding to \$w\$. The word \$w\$ is semiconcise if the subgroup \$[w(G),G]\$ is finite whenever \$G_w\$ is finite. The group \$G\$ is an \$FC(w)\$-group if the set of conjugates \$x^{G_w}\$ is finite for all \$xin G\$. We prove that if \$w\$ is a semiconcise word and \$G\$ is an \$FC(w)\$-group, then the subgroup \$[w(G),G]\$ is \$FC\$-embedded in \$G\$, that is, the intersection \$C_G(x)cap [w(G),G]\$ has finite index in \$[w(G),G]\$ for all \$xin G\$. A similar result holds for \$BFC(w)\$-groups, that are groups in which the sets \$x^{G_w}\$ are boundedly finite. We also show that this is no longer true if \$w\$ is not semiconcise.
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2020
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11386/4738742`
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