Inverse of a composition
Task number: 2441
Show that in each group holds \((a\circ b)^{-1}=b^{-1}\circ a^{-1}\).
Resolution
Oserve first that \((b^{-1}\circ a^{-1})\circ(a\circ b)= b^{-1}\circ (a^{-1} \circ a) \circ b= b^{-1}\circ e \circ b= b^{-1}\circ b= e\).
Hence both \((a\circ b)^{-1}\) and \(b^{-1}\circ a^{-1}\) are inverses of \(a\circ b\). Therefore they must be equal.
