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alg-07-35

alg-07-35 - 4.2 VII-35 SERIES OF GROUPS 51 4.2 VII-35...

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Unformatted text preview: 4.2. VII-35. SERIES OF GROUPS 51 4.2 VII-35. Series of Groups To give insights into the structure of a group G , we study a series of embed- ding subgroups of G . 4.2.1 Subnormal and Normal Series Def 4.10. A subnormal (or subinvariant) series of a group G is a finite sequence of subgroups of G : H = { e } < H 1 < H 2 < ··· < H n = G, such that H i C H i +1 (that is, H i is a normal subgroup of H i +1 ). Def 4.11. A normal (or invariant) series of a group G is a finite sequence of subgroups of G : H = { e } < H 1 < H 2 < ··· < H n = G, such that H i C G (that is, H i is a normal subgroup of G ). A normal series is always a subnormal series, but the inverse need not be true. If G is an abelian group, then every finite sequence of subgroups H = { e } < H 1 < H 2 < ··· < H n = G, is both a subnormal and a normal series. Ex 4.12 (Ex 35.2, p.311). Two examples of normal/subnormal series of Z under additions are { } < 8 Z < 4 Z < Z and { } < 9 Z < Z . Ex 4.13 (Ex 35.3, p.311). The group D 4 in Example 8.10. The series { ρ } < { ρ ,μ 1 } < { ρ ,ρ 2 ,μ 1 ,μ 2 } < D 4 is a subnormal series, but not a normal series. Def 4.14. 1 A subnormal/normal series { K j } is a refinement of a subnor- mal/normal series { H i } of a group G , if { H i } ⊆ { K j } , that is, if each H i is one of the...
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alg-07-35 - 4.2 VII-35 SERIES OF GROUPS 51 4.2 VII-35...

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