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{\centerline {\bf Homework 3 for 504, Fall 2018}}
{\centerline {due Wednesday, October 17 } }


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{\bf Problem 1.}  Let $G$ be  a $p$-group, and $H$ be a non-trivial normal subgroup of $G$. Show that $H \cap Z(G) \not = 1$. 
 
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{\bf Problem 2.} Let $p, q$ be prime numbers.
\begin{enumerate}
  \item Show that $\Aut(\Z/qZ) \simeq \Z/ (q-1)\Z$.
  \item Classify all groups of order $pq$.
  \end{enumerate}  

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Note: ``To classify" means to establish a list of representatives of each isomorphism class, and to prove that your list is complete. 

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{\bf Problem 3.}  (1).  Classify all finite groups of  orders  1 through 10.  \\
(2). Classify all groups of order 2015.

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{\bf Problem 4.}  (Sylow subgroups of symmetric groups) \\[3pt]
(1). What is the order of a Sylow 2-subgroup  of $S_{2^n}$?\\[3pt]
(2). Give an explicit description of a Sylow $2$-subgroup of  $S_{2^n}$.\\

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Note: You can think how this would work for  $S_{p^n}$, and, more generally, for any $S_n$. 

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{\bf Problem 5.}  Let $\F_p$ be a finite field of $p$ elements. 
%(Here, $p$ is a prime. The finite field $\F_p$ coincides with $\Z/p\Z$ as an abelian group. The non-zero elements in $\Z/p\Z$  also form a  group with respect to multiplication mod p. Hence, in $\F_p$ we can both add and multiply, and these operations are ``nicely" compatible. This allows to consider matrices over $\F_p$ the same way we consider matrices over $\CC$.)
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(1). Find the order of the finite group $\GL_n(\F_p)\stackrel{\rm def}{=}\{\text{invertible matrices over } \F_p \}$. \\[3pt]
(2). Let $U_n(\F_p)$ be the subgroup of $\GL_n(\F_p)$ consisting of matrices which have $1$'s on the main diagonal, $0$'s below, and arbitrary entries (from $\F_p$) above. Show that it is a Sylow p-subgroup in $\GL_n(\F_p)$. Is it unique? \\[3pt]
(3). Let $A, B$ be two commuting $n \times n$ matrices over $\F_p$  of order $p$ (that is, $A^p = B^p = \Id$ and $[A,B]=0$). 
Show that there exists a matrix $S \in \GL_n(\F_p)$ such that both $SAS^{-1}$ and $SBS^{-1}$ are upper-triangular (in other words, $A,B$ can be  simultaneously conjugated into upper-triangular form).   \\[3pt]
(4). Is this still true if $A$, $B$ do not commute? 




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