Normal subgroups of solvable groups

Posted: February 19, 2010 in Elementary Algebra; Problems & Solutions, Groups and Fields
Tags: , ,

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

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