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Journal of the London Mathematical Society (2)

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DERIVED SUBGROUPS OF PRODUCTS OF AN ABELIAN AND A CYCLIC SUBGROUP
M. D. E. CONDER, I. M. ISAACS
Journal:

Journal of the London Mathematical Society / Volume 69 / Issue 2 / April 2004

Published online by Cambridge University Press:

29 March 2004, pp. 333-348

Print publication:

April 2004
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Let $G$ be a finite group and suppose that $G = AB$, where $A$ and $B$ are abelian subgroups. By a theorem of Ito, the derived subgroup $G'$ is known to be abelian. If either of the subgroups $A$ or $B$ is cyclic, then more can be said. The paper shows, for example, that $G'{/}(G' \cap A)$ is isomorphic to a subgroup of $B$ in this case.

A STRUCTURAL PROPERTY CONCERNING ABSTRACT COMMENSURABILITY OF SUBGROUPS
R. I. GRIGORCHUK, J. S. WILSON
Journal:

Journal of the London Mathematical Society / Volume 68 / Issue 3 / December 2003

Published online by Cambridge University Press:

17 November 2003, pp. 671-682

Print publication:

December 2003
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Further properties of a group $\Gamma$ introduced by the first author in 1980 (sometimes called the first Grigorchuk group) are established. These are notably that any two infinite finitely generated subgroups of $\Gamma$ have isomorphic subgroups of finite index and that the generalized word problem for $\Gamma$ is soluble.