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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6320 • Algebra II

Fall 2021 • Question 2

Question

Math 6320 • Algebra II Fall 2021 • Question 2

Find a non-abelian finite subgroup of \(\operatorname {GL}_2(\mathbb {C})\) that is not conjugate to a finite subgroup of \(\operatorname {GL}_2(\mathbb {Z})\). Here for a ring \(R\), \(\operatorname {GL}_2(R)\) denotes the group of invertible \(2 \times 2\) matrices with entries in \(R\).

Raw LaTeX

Find a non-abelian finite subgroup of \(\operatorname{GL}_2(\mathbb{C})\) that
is not conjugate to a finite subgroup of \(\operatorname{GL}_2(\mathbb{Z})\).
Here for a ring \(R\), \(\operatorname{GL}_2(R)\) denotes the group of invertible
\(2 \times 2\) matrices with entries in \(R\).

Topic Tags

Finite Group Theory Normal Subgroups and Commutator Subgroups