## Normal subgroups of solvable groups

Posted: February 19, 2010 in Elementary Algebra; Problems & Solutions, Groups and Fields
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Problem. Let $G$ be a solvable group and $(1) \neq N \lhd G.$ Prove that there exists a non-trivial abelian subgroup $A$ of $N$ which is normal in $G.$

Solution. Let $(1) = G_0 \subset G_1 \subset \cdots \subset G_n = G$ be a normal series of $G$ and put

$k= \min \{j: \ N \cap G_j \neq (1) \}.$

Then $A=N \cap G_k$ is obviously normal in $G.$ Also, the map $f: A \longrightarrow G_k/G_{k-1}$ defined by $f(a) = aG_{k - 1}$ is an injective group homomorphism because $N \cap G_{k-1} = (1).$ Thus $A$ is abelian because $G_k/G_{k-1}$ is abelian. $\Box$