Representations of finite groups

In general, a representation of a finite group $𝐺$ with a finite carrier space is a homomorphism $Γ : 𝐺 \to S L (𝐺)$ , i.e. representation matrices of finite group elements have determinant 1. #m/thm/rep

Proof

Let $𝐺$ be a finite group and $Γ$ be a representation thereof with finite carrier space. Assume there exists $𝑔 \in 𝐺$ such that $| d e t Γ (𝑔) | = 𝑎 \neq 1$ . Then $| d e t Γ (𝑔 𝑛) | = | d e t (Γ (𝑔) 𝑛) | = 𝑎 𝑛 \neq 𝑎$ for all $𝑛 \neq 1$ , and therefore $𝑔 𝑛 \neq 𝑔$ for all $𝑛 \neq 1$ . Thus the cyclic subgroup $⟨ 𝑔 ⟩$ is infinite, contradicting our requirement. Therefore $| d e t Γ (𝑔) | = 1$ for all $𝑔 \in 𝐺$

#state/tidy | #lang/en | #SemBr