What style/font is this?

Partial answer that addresses:

• Dropped captital/number is solved by package lettrine. It also provides the small caps for Bezeichnungen:.

• Color of the number (gray).

• Alignment of the equation system (using environment alignat* of package amsmath. Just for fun I have added additional alignments for | and \in.

• Letter spacing of the small caps text via \textls of package microtype.

• As vertical bar \mid is used, see also question "\mid, | (vertical bar), \vert, \lvert, \rvert, \divides".

Also I have enlarged the number a bit for a better aligment with the top of the text.

\documentclass{article}
\usepackage[ngerman]{babel}
\usepackage[T1]{fontenc}
\usepackage[sc]{mathpazo}
\usepackage{tgpagella}
\usepackage{microtype}
\usepackage{amsmath}
\usepackage{lettrine}
\usepackage{xcolor}
\definecolor{theoremgray}{gray}{.5}

\setlength{\textwidth}{.85\textwidth}% just for reproducing the example

\begin{document}
\lettrine[
  loversize=.11,
  findent=0.5em,
  nindent=0pt,
]{\textcolor{theoremgray}{1.1.1}}{\textls{Bezeichnungen:}}
Ist $G$ eine (multiplikativ geschriebene) Gruppe, so f"uhrt man
f"ur beliebige $g \in G$ und $M, N \subseteq G$ die Bezeichnungen
\begin{alignat*}{4}
  &Mg &&= \{mg &&\mid m &&\in M\},\\
  &gN &&= \{gn &&\mid n &&\in N\},\\
  &MN &&= \{mn &&\mid m &&\in M, n \in N\}
\end{alignat*}
\end{document}

Missing:

• Wrapping the visual markup into proper logical markup by defining a theorem style and setting up a theorem, for example.

  The answer of Henri Menke shows this using package ntheorem as example.

Remarks

I reproduced Heiko's answer using ntheorem. There is a tricky thing about it: You can't wrap the \lettrine into the [] of the \item as one would normally do for theorems. I get the error

! Argument of \@lettrine has an extra }.

if I do this. I don't know why, maybe somebody knows (Please comment).

Implementation

\documentclass{article}
\usepackage[ngerman]{babel}
\usepackage[T1]{fontenc}
\usepackage[sc]{mathpazo}
\usepackage{tgpagella}
\usepackage{microtype}
\usepackage{amsmath}
\usepackage{lettrine}
\usepackage{xcolor}
\usepackage{ntheorem}
\definecolor{theoremgray}{gray}{.5}
\makeatletter
\newtheoremstyle{mytheorem}%
    {\item[]\theorem@headerfont \lettrine[loversize=.11,findent=0.5em,nindent=0pt]{\textcolor{theoremgray}{##2}}{\textls{##1\theorem@separator}} }%
    {\item[]\theorem@headerfont \lettrine[loversize=.11,findent=0.5em,nindent=0pt]{\textcolor{theoremgray}{##2}}{\textls{##3\theorem@separator}} }
\makeatother
\theoremstyle{mytheorem}
\theoremheaderfont{\normalfont}
\theorembodyfont{\normalfont}
\theoremseparator{:}
\newtheorem{theorem}{Theorem}[subsection]
\setlength{\textwidth}{.85\textwidth}% just for reproducing the example
\begin{document}
\section{Sectiontitle}
\subsection{Subsectiontitle}
\begin{theorem}[Bezeichnungen]
    Ist $G$ eine (multiplikativ geschriebene) Gruppe, so f"uhrt man
    f"ur beliebige $g \in G$ und $M, N \subseteq G$ die Bezeichnungen
    \begin{alignat*}{4}
      &Mg &&= \{mg &&\mid m &&\in M\},\\
      &gN &&= \{gn &&\mid n &&\in N\},\\
      &MN &&= \{mn &&\mid m &&\in M, n \in N\}
    \end{alignat*}
\end{theorem}
\end{document}

Output

Partial solution: the font is loaded with the package mathpazo.