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MAT $150 \mathrm{C}$, Spring 2021 Homework 1
Due before 12: 10 on Monday, April 5
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
$$x \mapsto\left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right)$$
Problem 3. 3. Consider the standard two-dimensional representation of the dihedral group $D_{n}$. For which $n$ is this an irreducible complex representation.