(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let [itex]G = GL_n(\mathbb{C})[/itex] and [itex]H = GL_n(\mathbb{R})[/itex]

2. Relevant equations

Prop. A subset H of a group G is a subgroup if:

1. If [itex]a,b \in H[/itex], then [itex]ab \in H[/itex].

2. [itex]1 \in H[/itex]

3. [itex]\forall a \in H[/itex] [itex]\exists a^{-1} \in H[/itex].

3. The attempt at a solution

My intuition says the answer isyessince the [itex]\mathbb{R} \subset \mathbb{C}[/itex], it seems reasonable to say the same about the general linear group of invertible [itex]n \times n[/itex] matrices. So if I am understanding how to check if H is a subgroup of G correctly, I need to test the three conditions above. Here goes:

1. Let [itex]a,b \in GL_n(\mathbb{R})[/itex]. Then the product of two [itex]n \times n[/itex] matrices [itex]a,b[/itex] is an [itex]n \times n[/itex] matrix [itex]ab[/itex]. Thus [itex]ab \in GL_n(\mathbb{R})[/itex].

2. The [itex]n \times n[/itex] identity matrix [itex]I \in GL_n(\mathbb{R})[/itex] since [itex]\det(I)=1[/itex] [itex]\forall n[/itex].

3. From the definition of [itex]GL_n(\mathbb{R}) = \{ n \times n \, \, \, invertible \, \, \, real \, \, \, matrices\}[/itex], it follows that [itex]\forall a \in GL_n(\mathbb{R})[/itex] [itex]\exists a^{-1} \in GL_n(\mathbb{R})[/itex].

So does this show that [itex]GL_n(\mathbb{R}) \subset GL_n(\mathbb{C})[/itex]? I'm confused since I haven't even mentioned [itex]GL_n(\mathbb{C})[/itex].

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# Checking if H is a subgroup of G

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