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\begin{problem}
Prove that if $P(x)$ and $Q(x)$ are polynomials over a field $F$ of degree
$m$ and $n$
respectively then the degree of $P(x)Q(x) = m + n$ while the degree of
$P(x) + Q(x) =
\operatorname{max}(m,n)$.
\end{problem}

\begin{problem} Let $G$ be a group and suppose that $G$ satisfies the
following property.
whenever $b\cdot a = a\cdot c$, then $b = c$ for $a,b,c \in G$. Prove that
$G$ must be
abelian.
\end{problem}

\begin{problem} Let $f: G \to H$ be an isomorphism of groups.  Prove that
$f(1_G) = 1_H$.
\end{problem}

\begin{problem} Kernel of a homomorphism is a subgroup.  Will we get to?
\end{proble}

\begin{problem} Let $p$ be a prime number and $G$ be a finite group of
order $p$.

a) Show that $G$ is cyclic.

b) Use part (a) to show that there is only one group of order $p$ up to
isomorphism.
\end{problem}

\begin{problem} How many {\em abelian} groups are there of order 24?
\end{problem}

\begin{problem}
Let $G = \Big{\{}
\begin{pmatrix}
a & a\\
a & a
\end{pmatrix}:
a \in \mathbb{R}^*\Big{\}}$.  $G$ is a group under matrix multiplication.

a) Find the identity element of this group.

b) Find the inverse of $\begin{pmatrix}
3 & 3\\
3 & 3
\end{pmatrix}$ in $G$.
\end{problem}

\begin{problem} The set $G =\{5, 15, 25, 35\}$ is a group under multiplication
modulo 40.

a) Find the identity element in $G$.

b) Find the inverses of the remaining elements in $G$.
\end{problem}

\begin{problem} Let $G$ be the set of rational numbers {\em except} $-1$. This
forms a group under the following multiplication: $a*b \equiv a + b + ab$,
where
the operations on the right hand side are as usual.

a) Find the identity element in $G$.

b) Find the inverse of 2 in $G$.
\end{problem}
\end{document}
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