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

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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
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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Categories: 数学代写数论