1993 November

Prove that there is no nonabelian simple group of order 36.

Prove that for all $n\geq 3$, the commutator subgroup of $S_n$ is $A_n$.

a. State, without proof, the Sylow Theorems.
b. Prove that every group of order 255 is cyclic. 
Let $G$ be a group and let $X$ be a set. We say that $G$ acts on $X$ (on the left) if there is a multiplication $G\times X \rightarrow X$ such that $1x = x$ and $(gh)x = g(hx)$ for all $g, h\in G$ and $x\in X$.
a. Let $H$ be a subgroup of $G$. Prove that $G$ acts on the space of left cosets $G/H$.
b. Let $x\in X$ and let $H$ be the subgroup of $G$ fixing $x$. Define the orbit of $x$ under $G$ and prove that there is a onetoone correspondence between this orbit and the space of left cosets $G/H$. In what sense can this onetoone correspondence be called an isomorphism?
c. Let $g\in G$ and let $x' = gx$. Let $H$ be the subgroup of $G$ fixing $x$ and let $H'$ be the subgroup fixing $x'$. Prove that $H$ and $H'$ are conjugate. 
Prove that a finite group is nilpotent if and only if it is a product of $p$groups.
{\bf Theorem.} [L. Sylow (1872)] Let $G$ be a finite group with $ G = p^m r$, where $m$ is a nonnegative integer and $r$ is a positive integer such that $p$ does not divide $r$. Then \begin{enumerate}[(i)] \item $G$ has a subgroup of order $p^m$. Such a subgroup is called a \emph{Sylow $p$subgroup} of $G$. \item If $H$ and $J$ are Sylow $p$subgroups of $G$, then $J\leq g H g^{1}$ for some $g\in G$. In particular, the Sylow $p$subgroups of $G$ form a single conjugacy class. \item Let $n_p$ denote the number of Sylow $p$subgroups of $G$ and let $H$ be any Sylow $p$subgroup of $G$. Then \[ n_p \equiv 1 \pmod p, \qquad n_p \mid [G:H], \quad \text{ and } \quad n_p = G : N_G(H). \] (Note that $[G:H]=r$ so the second condition says that $n_p$ divides $r$.) \end{enumerate}