# Conjugate elements in a group

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

• If you want to show that a statement is false, you should exhibit an example where the statement fails. Oct 24 '18 at 21:38

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 $$(12)(123)(12)^{-1} = (132)$$
• 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 $S_3$
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 $$T$$.