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Using generators to check for a normal subgroup

  1. Mar 31, 2013 #1
    1. The problem statement, all variables and given/known data
    Perhaps I should say first that this question stems from an attempt to show that in the group
    [itex] \langle x,y|x^7=y^3=1,yxy^{-1}=x^2 \rangle,\ \langle x \rangle [/itex] is a normal subgroup.

    Let [itex] G [/itex] be a group with a subgroup [itex] H [/itex]. Let [itex] G [/itex] be generated by [itex]A\subseteq G [/itex]. Suppose that [itex] H [/itex] is closed under conjugation by elements from [itex] A [/itex] in the sense that [itex] aha^{-1}\in H [/itex] for any [itex] a\in A [/itex] and any [itex] h\in H [/itex]. Is it then true that [itex] H [/itex] is a normal subgroup of [itex] G [/itex]?

    3. The attempt at a solution
    I know that [itex] G [/itex] is the set of all products of elements from [itex] A\cup A^{-1} [/itex] where [itex] A^{-1}:=\{a^{-1}|a\in A\}. [/itex] Since we are assuming that [itex] aha^{-1} \in H[/itex] for any [itex] a\in A [/itex] and any [itex] h\in H [/itex], it will be enough to show that [itex] a^{-1}ha\in H[/itex] for any [itex] a\in A [/itex] and any [itex] h\in H [/itex]. However I have not been able to show this.
  2. jcsd
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