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Do you know results about maximal normal subgroup among normal subgroups not containing a given element $x$ ? The problem can be reduce to the case of free groups.

First, such a sugroup exists thanks to Zorn lemma. Secondly, I think that if $x$ is a primitive element, then there is only one maximal normal subgroup among normal subgroups not containing $x$; otherwise, there is not unicity: if $x=[a,b]$, there is a one-to-one correspondance between our normal subgroup and non abelian two-generator groups whose proper quotients are abelian (eg. the dihedral group $D_3$ or the quaternion group $Q_8$).

But I have no idea about how construct such subgroups.

  • 0
    What is a primitive element?2012-07-16
  • 0
    A primitive element is an element of a free basis.2012-07-16

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