Abstract algebra不算是一门简单的学科，这门学科在国内叫做抽象代数，经常有很多学生在学linear algebra或者analysis(advance calculus)的时候觉得并不困难，但是却觉得Abstract algebra很难，这是因为没有找到正确的方法学习Abstract algebra，UpriviateTA有一系列非常擅长Abstract algebra的老师，可以确保您在Abstract algebra取得满意的成绩。
MAT $150 \mathrm{C}$, Spring 2021 Homework 1
Due before 12: 10 on Monday, April 5
Please write the homework solutions in connected sentences and explain your work. Mark the answers to each question. Scan or take pictures of your homework and upload it to Gradescope before due time.
以下是UC Davis的一次assignment.更多的经典案例请参阅以往案例，关于abstract algebra的更多的以往案例可以参阅相关文章。abstract algebra代写请认准UpriviateTA.
Problem 1. 1. Consider the cyclic group $G_{n}=\left\langle x \mid x^{n}=1\right\rangle$.
a) Describe all one-dimensional complex representations of $G_{n}$.
b) Prove that every complex representation of $G_{n}$ has a one-dimensional invariant subspace.
a) Describe all one-dimensional complex representations of $G_{n}$.
b) Prove that every complex representation of $G_{n}$ has a one-dimensional invariant subspace.
Problem 2. 2. a) Prove that there is a two-dimensional representation of $G_{4}$ such that
$$
x \mapsto\left(\begin{array}{cc}
0 & 1 \\
-1 & 0
\end{array}\right)
$$
b) Find all invariant subspaces for the corresponding real representation.
c) Find all invariant subspaces for the corresponding complex representation.
$$
x \mapsto\left(\begin{array}{cc}
0 & 1 \\
-1 & 0
\end{array}\right)
$$
b) Find all invariant subspaces for the corresponding real representation.
c) Find all invariant subspaces for the corresponding complex representation.
Problem 3. 3. Consider the standard two-dimensional representation of the dihedral group $D_{n}$. For which $n$ is this an irreducible complex representation.
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