Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2007 • Question 6

Question

Math 6310 • Algebra Spring 2007 • Question 6

Let \(N\) be a normal subgroup of a finite group \(G\). Assume that the center of \(N\) is trivial and that any automorphism of \(N\) is inner. Show that there exists a normal subgroup \(H\) of \(G\) such that \(G\) is isomorphic to the direct product of \(H\) and \(N\).

Raw LaTeX

Let \(N\) be a normal subgroup of a finite group \(G\). Assume that the center of \(N\) is trivial
and that any automorphism of \(N\) is inner. Show that there exists a normal subgroup \(H\) of \(G\)
such that \(G\) is isomorphic to the direct product of \(H\) and \(N\).

Topic Tags

Chinese Remainder Theorem and Product Decompositions Finite Group Theory Normal Subgroups and Commutator Subgroups Tensor Products of Modules