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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2018 • Question 2

Question

Math 6310 • Algebra Fall 2018 • Question 2

Let \(G\) be a finite simple group with identity \(e\). Suppose that \(A\) and \(B\) are distinct maximal proper subgroups of \(G\). If \(A\) and \(B\) are abelian, prove that \(A\cap B=\{e\}\).

Hint: Prove that \(A\cap B\) is normal in \(G\).

Raw LaTeX

Let \(G\) be a finite simple group with identity \(e\). Suppose that \(A\) and \(B\) are distinct maximal
proper subgroups of \(G\). If \(A\) and \(B\) are abelian, prove that \(A\cap B=\{e\}\).

\emph{Hint:} Prove that \(A\cap B\) is normal in \(G\).

Topic Tags

Abelian Groups and Classification of Finite Abelian Groups Finite Group Theory Normal Subgroups and Commutator Subgroups Simple Groups