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I see that normal subgroups are closed w.r.t conjugates (Pinter, A book of abstract algebra, pg. 140).

I would like to ask what are some of the algebraic properties shared by conjugates ($xax^{-1} $ where $x \in G$) of an element $a$ in a group $G$ ? For instance I could observe that conjugate elements share the same order.

And also, how can such properties be derived. For instance how can I show that the order of an element and its conjugate is the same?

I guess this might help in giving an idea about the type of elements contained in a normal subgroup.

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If $b=gag^{-1}$ and $a^n=1$, then $b^n=1$.

Therefore, the set of exponents that kill $a$ is contained in the set of exponents that kill $b$.

Since $a=g^{-1}bg$, we have the reverse inclusion and $a,b$ have the same set of exponents. In particular, their order is the same, being the minimum of the set.

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