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How to prove determinant is a group homomorphism and onto?.

I posted this question I am struggling with previously but it was put on hold for lack of context, I hope this is now clearer. Consider the determinant function Det: Mn($\mathcal{F}$) $\to$ ...
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I have to check, weather the following subset of a group is also a subgroup: $$U = \left\{ \begin{pmatrix} a & -b \\ \overline{b} & \overline{a} \end{pmatrix} \in GL(2, \mathbb{C}) \bigg\vert ... 1answer 40 views What is the order of this group? [duplicate] Let H be the subgroup of the group G of all 2 \times 2 non-singular matrices whose entries are integers modulo a given prime p consisting of those and only those matrices in G whose ... 1answer 3k views Proof relation between Levi-Civita symbol and Kronecker deltas in Group Theory In order to prove the following identity:$$\sum_{k}\epsilon_{ijk}\epsilon_{lmk}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$$Instead of checking this by brute force, Landau writes thr product of ... 3answers 153 views Is there a way to get all the permutations of S_4 I need to calculate the determinant of a 4 \times 4 matrix by "direct computation", so I thought that means using the formula$$\sum_{\sigma \in S_4} (-1)^{\sigma}a_{1\sigma(1)}\ldots ...
Consider the functions $\rho_1:M_1(\mathbb C)\to M_2(\mathbb R)$ where $$\rho_1(a+bi)=\begin{pmatrix} a&b\\ -b&a \end{pmatrix}$$ and $\rho_2:M_2(\mathbb C)\to M_4(\mathbb R)$ where ...
Help deriving that $\mathrm{sign} : S_n\to \{\pm 1\}$ is multiplicative
$\def\sign{\operatorname{sign}}$ For homework, I am trying to show that $\sign:S_n \to \{\pm 1\}$ is multiplicative, i.e. that for any permutations $\sigma_1,\sigma_2$ we have \sign(\sigma_1 ...