課程名稱︰代數一
課程性質︰數學系選修 數學系碩士班數學組必選
課程教師︰朱樺
開課系所︰數學系所
考試時間︰950114
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
(1) (20%) The derived series of a group are the groups $G^{(1)}=G'$,
$G^{(i+1)}=(G^{(i)})'$. Find the derived series of the group $S_n$, $n>2$.
(2) (20%) Let $SM$ denote the group of orientation-preserving motions of
the real plane. Show that $SM$ is isomorphic to the subgroup
$\{\begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix}: |a|=1, b\in \mathbb{C}\}$
of $\GL_2(\mathbb{C})$.
(3) (20%) Let $G$ be a finite group and let $n<|G|$. Show that $G$ is
isomorphic to a transitive subgroup of $S_n$ if and only if $G$ contains
a subgroup $H$ of index $n$ such that neither $H$ nor any proper subgroup
of $H$ is normal in $G$.
(4) (20%) Analyze the group generated by $x, y$, with relations
$x^4=x^2y^2=(xy)^2=1$.
(5) (20%) the Lorentz group is defined to be the group
$O_{3,1}=\{P \in M_4(\mathbb{R}): P^tI_{3,1}P=I_{3,1}\}$,
where $I_{3,1}=\begin{pmatrix} I_3 & \\ & -1 \end{pmatrix}$.
(a) Show that $e^{tA}$ is a one -parameter subgroup of $O_{3,1}$
if and only if $I+\epsilonA \in O_{3,1}$ where $\epsilon$ is a
formal infinitesimal element.
(b) Find the Lie algebra $\Lie(O_{3,1})$ of the group $O_{3,1}$.
Find the dimension of $\Lie(O_{3,1})$ over $\mathbb{R}$.
(6) (25%) Classify all groups of order 36.
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