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Show that a given general linear group is nonabelian


Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.3

Show that $GL_2(\mathbb{F}_2)$ is non-abelian.


Solution: We have $$\left[ {1 \atop 1}{1 \atop 0} \right] \cdot \left[ {0 \atop 1}{1 \atop 0} \right] = \left[ {1 \atop 0}{1 \atop 1} \right]$$ and $$\left[ {0 \atop 1}{1 \atop 0} \right] \cdot \left[ {1 \atop 1}{1 \atop 0} \right] = \left[ {1 \atop 1}{0 \atop 1} \right],$$ so $GL_2(\mathbb{F}_2)$ is non-abelian.


Linearity

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