Tagged: group

Special Linear Group is a Normal Subgroup of General Linear Group

Problem 332

Let $G=\GL(n, \R)$ be the general linear group of degree $n$, that is, the group of all $n\times n$ invertible matrices.
Consider the subset of $G$ defined by
\[\SL(n, \R)=\{X\in \GL(n,\R) \mid \det(X)=1\}.\] Prove that $\SL(n, \R)$ is a subgroup of $G$. Furthermore, prove that $\SL(n,\R)$ is a normal subgroup of $G$.
The subgroup $\SL(n,\R)$ is called special linear group

 
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The Additive Group $\R$ is Isomorphic to the Multiplicative Group $\R^{+}$ by Exponent Function

Problem 322

Let $\R=(\R, +)$ be the additive group of real numbers and let $\R^{\times}=(\R\setminus\{0\}, \cdot)$ be the multiplicative group of real numbers.

(a) Prove that the map $\exp:\R \to \R^{\times}$ defined by
\[\exp(x)=e^x\] is an injective group homomorphism.

(b) Prove that the additive group $\R$ is isomorphic to the multiplicative group
\[\R^{+}=\{x \in \R \mid x > 0\}.\]

 
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