# Character table for cyclic group of order 7

1. Feb 23, 2014

### CAF123

1. The problem statement, all variables and given/known data
a)Write down all irreducible representations of $\mathbb{Z}_7$.
b)How many of the irreducible representations are faithful?

2. Relevant equations
Group structure of $\mathbb{Z}_7 = \left\{e^{2\pi in/7}, \cdot \right\}$ for $n \in \left\{0,....,6\right\}$

3. The attempt at a solution
I am trying to construct the character table of $\mathbb{Z}_7$. Since this is an abelian group, there are 7 conjugacy classes and thus 7 one dimensional irreducible representations. Since the representations are all 1D, the character of the representation is simply the representation itself. My question is are the representations of the form $e^{2\pi in/7}?$ Letting n run from 0 to 6 does indeed give 7 irreducible representations.

I have in some notes the following statement: Let $\rho$ denote a representation. Then $$\rho(1)^7 = \rho(1^7) = \rho(1) = 1$$ Since $\rho$ is always 1D, it can be written as a complex scalar. Let $x = \rho(1) \Rightarrow x^7 - 1 = 0$ and thus the seven irreducible reps of 1 are of the form $e^{2\pi in/7}$ Do I do a similar analysis for the other elements? I wasn't sure though why $\rho(1^7) = \rho(1)$ though. $1 \in \mathbb{Z}_7 = \left\{n, + | n \in \left\{0,...,6\right\}\right\}$, so I think $1^7 = 1 + 1...+1 = 0$ mod$7$, by summing 1 seven times.

To mentors: This is from a course on symmetries of QM, so that is why I posted it here. Perhaps it is better elsewhere.

Thanks.