Question

prove that a group G is solvable if and only if G(k) = (e) for some integer K

Answer 1

"Let G be a group.

G = H0 ⊇ H1 ⊇ • • • Hn−1 ⊇ Hn = {e}

of subgroups is called a solvable series for G if Hi+1 is normal in Hi and Hi/Hi+1 is commutative

for every i = 0, 1, • • • , n − 1.

A group G is called a solvable group if G has a solvable series.

Every abelian group is solvable. For, if G is abelian, then G = H0 ⊇ H1 = {e} is a solvable

series for G.

"