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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 1995 • Question 9

Question

Math 6310 • Algebra Fall 1995 • Question 9

Assume that \(A\) and \(B\) are solvable subgroups of the group \(G\) and \(G=AB\). Here \(AB=\{ab\mid a\in A,\ b\in B\}\).

  1. Give an example showing that \(G\) does not have to be solvable.
  2. Prove that if either \(A\) or \(B\) is a normal subgroup of \(G\), then \(G\) is solvable.

Raw LaTeX

Assume that \(A\) and \(B\) are solvable subgroups of the group \(G\) and \(G=AB\). Here
\(AB=\{ab\mid a\in A,\ b\in B\}\).
\begin{enumerate}
\item[(a)] Give an example showing that \(G\) does not have to be solvable.
\item[(b)] Prove that if either \(A\) or \(B\) is a normal subgroup of \(G\), then \(G\) is solvable.
\end{enumerate}

Topic Tags

Finite Group Theory Normal Subgroups and Commutator Subgroups Solvable Groups