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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 12.2  \\ Spring 2019 \hfill Due May 1}}}

\large

\vspace{1cm}


\begin{exercise}
Let $G$ be a group, $S$ a set, and 
\[
\begin{split}
G &\rightarrow \operatorname{Perm}(S),\\
x & \longmapsto \alpha_{x},
\end{split}
\]
a homomorphism.  For $x \in G$ and $s \in S$, define
\[
x \cdot s = \alpha_x(s).
\]
Show that this is an action of $G$ on $S$.
\end{exercise}

\begin{exercise}  
Let $G = \operatorname{SL}_2(\R) = \{ A \in \operatorname{GL}_2(\R) \, | \, \det A = 1 \}$ and $\mathcal{H} = \{ z \in \C \, | \, \operatorname{Im} z > 0 \}$.  For $\gamma = \left( \begin{smallmatrix}  a & b \\ c & d  \end{smallmatrix} \right) \in G$, and $z \in \mathcal{H}$, prove that
\[
\gamma z = \frac{az+b}{cz+d}
\]
is a well-defined action of $G$ on $\mathcal{H}$.  [{\em Suggestion:}  The ``hard'' part is proving associativity of the action.  This can certainly be done directly, but it can also be facilitated by working on $\mathbb{P}^1(\C)$, the (projective) Riemann sphere, instead of $\mathcal{H}$.]
\end{exercise}

\begin{exercise}
Let $G$ be a group acting on a set $S$.  Any subgroup of $G$ acts on $S$ by restriction, by what about factors of $G$?  Let $N \lhd G$.  The natural way to let $G/N$ act on $S$ would be via representatives.  That is, we would like to define $(xN) \cdot s = xs$ for $xN \in G/N$ and $s \in S$.  But in order for this to be well-defined, we must have $xs=ys$ when $xN = yN$ and $s \in S$, which may not happen.

Prove that if we replace $S$ by the subset $S^N = \{s \in S \, | \, ns = s \text{ for all } n \in N \}$ of {\em $N$-invariants} of $S$, then the rule above yields a well-defined action of $G/N$ on $S^N$.\footnote{This result is used in algebraic number theory, when trying to define the Euler factor at a ramified prime for the Artin $L$-function of a finite dimensional representation of the group of a Galois extension of number fields.}  Moreover, show that if $G/N$ acts ``naturally'' on any other $T \subset S$, then $T \subset S^N$.
\end{exercise}


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