In finite group we usually construct a composition series by a sequence of maximal normal subgroups. Can we construct a composition series in a finite group by a sequence of maximal normal subgroups ? Let G be a finite group. If G is simple. Then we are done. If G is not simple then we get a minimal normal subgroup of G. Now we take the quotient group and the quotient group is simple then we are done. If the quotient is not simple then we get a minimal normal subgroup of that and repeat the same process. At some point it will stop since G is finite. Now the question is : will this produce a composition series or not ? If not, Why ?
2026-03-28 05:00:51.1774674051
Composition Series in a finite group
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"Now if H is a minimal normal subgroup of G then the quotient G/H is also simple." is wrong. Example: $C_2\times C_2\times C_2$.
$C_2$ is a minimal normal and the quotient is $C_2\times C_2$, not simple.