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Basic Abstract Algebra by S. K. Jain, S. R. Nagpaul, P. B. Bhattacharya

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CHAPTER 6

Normal series

1 Normal series

Definition. A sequence (G0,G1, …,Gr) of subgroups of a group G is called a normal series (or subnormal series,) of G if

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The factors of a normal series are the quotient groups Gi /Gi–1, 1 ≤ ir.

Definition. A composition series of a group G is a normal series (G0, …,Gr) without repetition whose factors Gi/Gi-1 are all simple groups. The factors Gi/Gi-1 are called composition factors of G.

We often refer to a normal series (G0, G1, …,Gr) by saying that

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is a normal series of G.

For any group is trivially ...

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