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\def\GL{\operatorname {GL}}
\def\RR{{\mathbb R}}
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{\centerline {\bf Homework 2 for 504, Fall 2018}}
{\centerline {due  Wednesday, October 10, }}


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\noindent
{\bf Problem 0.} Prove that ${ p^am \choose p^a}$ is not divisible by $p$, a prime,  if $(m,p)=1$, and $a$ is a positive integer. 


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{\bf Problem 1.} 
\begin{enumerate}
  \item Prove the Third Isomorphism Theorem (all three parts of it): 
  \begin{enumerate}
  \item Let $H \triangleleft G$ be a normal subgroup of $G$. Show that there is one-to-one correspondence between subgroups of $G$ containing $H$ and subgroups of $G/H$
  \item Show that normal subgroups correspond to normals subgroups under this correspondence
  \item Let $K\lhd G, H \lhd G$ be  {\it normal} subgroups of $G$ fitting into a tower $K \lhd H \lhd G$. Show that there is an isomorphism 
  \[G/H \cong \frac{G/K}{H/K}. \] 
  \end{enumerate}  
  \item  Give an example of a tower of groups $K \lhd H \lhd G$ such that $K$ is not normal in $G$.
 \end{enumerate}  

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{\bf Problem 2.}  
Let $H <G$ be a subgroup. Show that the normalizer $N_G(H) = \{g \in G \, | \, gHg^{-1} =H \}$  is the maximal subgroup of $G$ in which $H$ is normal. 


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{\bf Problem 3.} Let $H$ be a subgroup of $G$ of index 2. Show that $H$ is normal. 

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{\bf Problem 4.} Let $G$ be a finite group, and $H \lhd G$ be a normal subgroup. Suppose $(|H|, [G:H]) =1$.
Show that $H$ is the  ONLY subgroup of $G$ of order $|H|$. 

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\noindent
{\bf Definition.} Let $G$ be a group acting on the set $X$. 
 The subgroup 
$$G_x = \rm {Stab}_G(x) = \{ g \in G \, | \, gx =x \}$$
 is called the {\bf stabilizer} or the {\bf isotropy group} of $x$.

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\noindent
{\bf Problem 5.}   
(1). Consider the ``standard" action of $\GL_n(\RR)$  on $\RR^n$ (that is, an $n \times n$ invertible matrix acts on an $n \times 1$ vector by matrix multiplication). Describe the orbits and  isotropy groups (up to conjugation) of this action. \\
(2). Let $\GL_n(\CC)$ act on the set of all $n \times n$ complex matrices by conjugation. Determine the orbits.


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\noindent
{\bf Problem 6.}
Show that any group of order $p^2$ for $p$ prime  is abelian.


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