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Suppose H is a subgroup of G, and h and h' are elements in H. How could I show this is true or false: If h and h' are conjugates in G, then they are conjugates in H.

My attempt:suppose h and h' are conjugates in G then there exists and k$\in$G such that h=kh'$k^{-1}$ but this k may not always belong in H so this seems that it may be false. How could I show this? thanks

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    $\begingroup$ If you want to show that a statement is false, you should exhibit an example where the statement fails. $\endgroup$
    – Lucas
    Oct 24 '18 at 21:38
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In general it is false, just consider the cases in which $H$ is abelian when two different elements cannot be conjugates. For a more concrete example let us consider $S_3$ the group of permutation of 3 elements. Take $H = <(123)>$, since it is abelian the elements $(123)$ and $(132)$ are not conjugates in $H$ however $(12)(123)(12)^{-1} = (132)$

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  • $\begingroup$ what do you mean by <(123)> is abelian? I though if n greater than 2 then Sn not abelian $\endgroup$
    – Rivaldo
    Oct 24 '18 at 22:11
  • $\begingroup$ does this make sense? : H=(1 2 ) abelian group so (1 2) and (2 1) can't be conjugates in H however (2)(21)(1)=(12) $\endgroup$
    – Rivaldo
    Oct 24 '18 at 22:17
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    $\begingroup$ By <(123)> I meant the sub group generated by (123) i.e. $H={Id, (123), (132) }$, be careful that the permutation $(12)$ and $(21)$ are the same elements in $S_3$ $\endgroup$
    – ALG
    Oct 25 '18 at 7:18
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Consider the affine group of dimension $n$, let $T$ be the subgroup of translations, and $A$ an invertible matrix distinct of the identity, let $f$ a translation, $AfA^{-1}$ is a translation, and $f$ is not conjugated with $AfA^{-1}$ in $T$.

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