Olympiad problems

1993 Hungary-Israel Binational, 2

In the questions below: $G$ is a ﬁnite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group. Suppose that $n \geq 1$ is such that the mapping $x \mapsto x^{n}$ from $G$ to itself is an isomorphism. Prove that for each $a \in G, a^{n-1}\in Z (G).$

1951 Miklós Schweitzer, 14

Tags: group theory , abstract algebra , superior algebra , superior algebra unsolved

For which commutative finite groups is the product of all elements equal to the unit element?

2013 Miklós Schweitzer, 3

Tags: group theory , abstract algebra , superior algebra , superior algebra unsolved

Find for which positive integers $n$ the $A_n$ alternating group has a permutation which is contained in exactly one $2$-Sylow subgroup of $A_n$. [i]Proposed by Péter Pál Pálfy[/i]