WJD Open Notebook

1993 November

1. Prove that there is no non-abelian simple group of order 36.

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

3. a. State, without proof, the Sylow Theorems.
   b. Prove that every group of order 255 is cyclic.

4. 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 one-to-one correspondence between this orbit and the space of left cosets $G/H$. In what sense can this one-to-one 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.

5. 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 non-negative 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}