On locally finite groups with a four-subgroup whose centralizer is small. (Q372817)
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scientific article; zbMATH DE number 6217357
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On locally finite groups with a four-subgroup whose centralizer is small. |
scientific article; zbMATH DE number 6217357 |
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On locally finite groups with a four-subgroup whose centralizer is small. (English)
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21 October 2013
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The main result of this paper is: Let \(G\) be a locally finite group which contains a Klein four-subgroup \(V\) such that \(C_G(V)\) is finite and \(C_G(v)\) has finite exponent for some \(v\in V\). Then \([G,v]'\) has finite exponent. As a corollary it is proved that, under the previous hypothesis, the group \(G\) has a normal series \[ \{1\}\leq G_1\leq G_2\leq G_3\leq G \] such that \(G_1\) and \(G/G_2\) have finite exponents while \(G_2/G_1\) is Abelian. Moreover \(G_3\) is hyperabelian and has finite index in \(G\).
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locally finite groups
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centralizers
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Klein four group
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groups of finite exponent
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