(joint work with O.~Bogopolski and H.~Zieschang)

\medskip In~1930 Magnus proved that {\sl if elements $u,v$ of a free group $\cal F$ have the same normal closures, then $u$ is conjugate to $v^{\pm1}$}. We know the following two generalizations of this theorem.

1) In~1961 Greendlinger proved that if two subsets $\cal U$ and $\cal V$ of a free group $\cal F$ satisfy some small cancellation conditions and have the same normal closures then there is a bijection $\psi\colon\,{\cal U}\rightarrow \cal V$ such that $u$ is conjugate to $\psi(u)^{\pm1}$.

2) A group is said to be locally indicable if each of its non-trivial, finitely generated subgroups admits an epimorphism onto the infinite cyclic group. Let $\cal A$ and $\cal B$ be two non-trivial locally indicable groups. In~1989 Edjvet proved that if $u,v\in {\cal A}\ast \cal B$ are cyclically reduced words each of length at least two, and if the normal closures of $u$ and $v$ coincide, then $u$ is a conjugate of $v^{\pm1}$.

\smallskip\noindent {\bf Theorem 1.}~{\sl Let $S$ be a closed surface and $g,\,h$ non-trivial elements of $\pi_1(S)$ both containing simple closed two-sided curves $\gamma$ and $\chi$, resp. If $h$ belongs to the normal closure of $g$ then $h$ is conjugate to $g^\varepsilon$ or to $(gug^{\eta}u^{-1})^\varepsilon$, $\varepsilon,\eta \in \{1,-1\}$; here $u$ is a homotopy class containing a simple closed curve $\mu$ which properly intersects $\gamma$ exactly once.

Moreover, if $h$ is not conjugate to $g^\varepsilon$ then $\eta = 1$ if $\mu$ is one-sided and $\eta = -1$ elsewise, and $\chi$ is homotopic to the boundary of a regular neighbourhood of $\gamma\cup\mu$.}

\medskip A direct consequence is the following analog of the Magnus' theorem.

\smallskip\noindent {\bf Corollary.}~{\sl Let $S$ be a closed surface and $g,\,h$ be non-trivial elements of $\pi_1(S)$ both containing simple closed two-sided curves. If the normal closures of $g$ and $h$ coincide then $h$ is conjugate to $g$ or $g^{-1}$.}

\medskip The proof of Theorem~1 is geometrical and uses coverings, intersection numbers of curves, and Brouwer's fixed-point theorem. As a corollary we obtain the following Theorem~2 concerning {\it normal} automorphisms (an automorphism of a group $\cal G$ is called normal if it maps each normal subgroup of $\cal G$ into itself).

\smallskip\noindent {\bf Theorem 2.}~\cite{BKZ}~{\sl If $S$ is a closed surface different from the torus and the Klein bottle, then every normal automorphism of $\pi_1(S)$ is an inner automorphism.}

\medskip Earlier Lubotsky~(1980) and Lue~(1980) proved that every normal automorphism of a free group of rank at least 2 is an inner automorphism. In~1996 Neshchadim proved that any normal automorphism of the free product of two non-trivial groups is an inner automorphism.

Remark that Theorem~1 and Corollary admit some analogs for non-simple curves satisfying additional assumptions.