Show that <a, b|a^5=b^2=e, ba=(a^2)b> is isomorphic to Z2.
Let the game begin... :
$\displaystyle bab^{-1}=a^2\Longrightarrow ba^2b^{-1}=\left(bab^{-1}\right)^2=a^4$ , but also $\displaystyle ba^2b^{-1}=b\left(bab^{-1}\right)b^{-1}=b^2ab^{-2}=a$ , so we get $\displaystyle a=a^4\Longrightarrow a^3=1$ , which together with the given data $\displaystyle a^5=1$
means that $\displaystyle a=1$ ,and thus in fact $\displaystyle \left<a,b\;/\;a^5=b^2=1\,,\,ba=a^2b\right>=\left<b\;/\;b^2=1\right>\cong\mathbb{Z}_2$
Tonio
The first part of your statement/question is right, but the second part is not true `because of the definition of conjugation'. Tonio has substituted in $\displaystyle a^2 = bab^{-1}$ which can be done as $\displaystyle ba=a^2b \Rightarrow bab^{-1} = a^2$.
How about...$\displaystyle baba=a^2a=a^3$ (as $\displaystyle bab = a^2$ by $\displaystyle ba=a^2b$). This gives us that $\displaystyle bab=1$ by cancelling the $\displaystyle a$s and so we have that $\displaystyle <a, b; a^5 = b^2=1, ba=a^2b> = <b; b^2>$ as required.