## Quiz description

You should show all of your calculations and explain what you are doing.
You may use a calculator to do arithmetic, but the use of software packages like Mathematica, Sage, etc to solve problems is not allowed

After you have completed the quiz, please save, scan, or take photos of your work and upload your files to the questions below. Crowdmark accepts PDF, JPG, and PNG file formats.

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.
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.
Problem 3. 3. Consider the standard two-dimensional representation of the dihedral group $D_{n}$. For which $n$ is this an irreducible complex representation.

Categories: 数学代写数论