Question

Q.1 Define normal and subnormal series with example.​

Answer 1

Answer:

If G is nilpotent then the ascending central series C1(G) < C2(G) < ··· < Cn(G) = G is a normal series for G since each Ci(G) is normal in G. Definition II. ... A subnormal series G = G0 > G1 > ··· > Gn = {e} is a composition series if each factor Gi/Gi+1 is simple.

Answer 2

If G is nilpotent, the ascending central series 〈e〉 < C1(G) < ··· < Cn(G) = G is a normal series of G. Def. A subnormal series G = H0 > H1 > ··· > Hm is a refinement of a subnormal series G = G0 > G1 > ··· > Gn if G0, G1, ··· , Gn is a subsequence of H0, H1, ···, Hm

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