Let $N(n)$ denote the smallest positive integer $N$ such that $x^N =e$ for every element $x$ of the symmetric group $S_n$, where $e$ denotes the identity permutation. Prove that if $n>1,$ $$\frac{N(n)}{N(n-1)} =\begin{cases} p \;\text{if}\; n\; \text{is a power of a prime } p\\ 1\; \text{otherwise}. \end{cases}$$

Let be a natural number $ n\ge 2, $ a real number $ \lambda , $ and let be the set $$ H_{\lambda }=\left\{ \left( h_k^l \right)_{1\le k\le n}^{1\le l\le n}\in\mathcal{M}_n\left(\mathbb{R}\right) \bigg| \lambda =\sum_{k,l=1}^n h_k^l \right\} . $$ Prove the following statements. [b]a)[/b] The sets of symmetric and antisymmetric matrices from $ \mathcal{M}_n\left(\mathbb{R}\right) $ are subgroups of the additive subgroup $ \mathcal{M}_n\left(\mathbb{R}\right) , $ and any matrix from $ \mathcal{M}_n\left(\mathbb{R}\right) $ is a sum of a symmetric and antisymmetric matrix from $ \mathcal{M}_n\left(\mathbb{R}\right) . $ [b]b)[/b] $ \left( H_{\lambda },+\right)\le\left( \mathcal{M}_n\left(\mathbb{R}\right) ,+ \right)\iff \lambda =0 $ [b]c)[/b] There is a commutative group formed with the elements of $ H_{\lambda } $ if $ \lambda\neq 0. $ [i]Dan Negulescu[/i]