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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 1997 • Question 8

Question

Math 6310 • Algebra Fall 1997 • Question 8

Let \(A_n\) be the alternating group of degree \(n\) (i.e., \(n\) is the number of objects permuted by \(A_n\)). If \(n\ge 5\) and \(H\) is a subgroup of \(A_n\) such that the order of \(H\) is \(\frac {(n-1)!}{2}\), then prove that \(H\) is isomorphic to \(A_{n-1}\).

Raw LaTeX

Let \(A_n\) be the alternating group of degree \(n\) (i.e., \(n\) is the number of objects permuted by
\(A_n\)). If \(n\ge 5\) and \(H\) is a subgroup of \(A_n\) such that the order of \(H\) is
\(\frac{(n-1)!}{2}\), then prove that \(H\) is isomorphic to \(A_{n-1}\).

Topic Tags

Finite Group Theory Ideals and Quotient Rings Normal Subgroups and Commutator Subgroups Symmetric and Alternating Groups