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\title{Group Theory: Math30038, Sheet 3}
\author{GCS}
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{\it The course web site is available via
{\tt http://www.bath.ac.uk/$\sim$masgcs/}}

\begin{enumerate}
\item Let $\phi$ denote the Euler $\phi$-function.
Prove that for every integer $n$ we have
\[ \sum \phi(d) = n\]
where the sum is taken over all natural numbers $d$ which
divide $n$.
\item Suppose that $G$ is a finite abelian group. Suppose
that $p$ is a prime number which divides $|G|$. Prove that
there is an element $g \in G$ such that $o(g) = p$.
{\em Hint: multiply together all the cyclic subgroups of $G$.}
\item Suppose that $G$ is a group and that $K$, $L$ are both
normal subgroups with the property that $K \cap L = 1$ (i.e.
$K$ and $L$ intersect to form the trivial subgroup consisting 
of the identity element). Prove that every element of $K$
commutes with every element of $L$. {\em Hint: consider elements
of the form $k^{-1}l^{-1}kl$ where $k \in K$ and $l \in L$.}
\item Suppose that $G$ is a group and that $H$ is a subgroup
of $G$ of finite index. Suppose that $K$ is also a subgroup of $G$.
Prove that $\vert K : H \cap K \vert \leq \vert G : H \vert.$
What can you say if this inequality is an equality? 
\item Let $G$ be a group. Suppose that $H \leq G$ and that
$\vert G : H \vert = 2$. Prove that $H \unlhd G$. Can one derive
the same conclusion when 2 is replaced by 3?
\end{enumerate}
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