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Groups, Rings and Fields [Lecture notes] by Karl-Heinz Fieseler PDF

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By Karl-Heinz Fieseler

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Extra resources for Groups, Rings and Fields [Lecture notes]

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Is a free generator system of an abelian group F with a = b and g, h ∈ G arbitrary elements in some group G, there is a group homomorphism ϕˆ : F −→ G with ϕ(a) ˆ = g, ϕ(b) ˆ = h. But then ab = ba implies gh = hg, so we could conclude that any group is abelian! 4. ). If a ∈ M , there is a group homomorphism ϕˆ : F −→ R∗ with ϕ(a) ˆ = 2. Since 2 ∈ R∗ has infinite order, the element a ∈ F has as well. 57. Let Fi ⊃ M be groups freely generated by M for i = 1, 2 and ϕi : M → Fi the inclusion. Then ϕˆ2 : F1 −→ F2 is an isomorphism.

R: Let H ⊂ Z2 be the subgroup generated by (a, b), (c, d) ∈ Z. 84! 3. Show that Aut(Zn ) ∼ = GLn (Z) := {A ∈ Zn,n , det(A) = ±1}. Furthermore for n,n A ∈ Z , det A = 0, that the index of the subgroup A(Zn ) ⊂ Zn is | det A|. 4. R: Write the groups Z∗n , cf. 84! 85 induces an isomorphism Z∗q −→ Z∗k1 × ... × Z∗pkr for q = pk11 · ... · pkr r . p1 r ∼ 5. Show: = Z2 and Z∗2n ∼ = Z2 × Z2n−2 for n > 2. More precisely ∗ ∼ Z2n =< −1 > × < 1 + 4r > with any odd number r ∈ Z. Z∗4 6. e. such that F −→ G −→ G/T (G) is an isomorphism.

E. the homomorphism σ : F −→ Aut(H), f → κf |H . If σ ≡ idH , then G ∼ = H × F . 48. Let F, H be groups and σ : F −→ Aut(H), f → σf := σ(f ) a group homomorphism. Then the group H ×σ F := (H × F, µ := µσ ), where µσ ((h, f ), (h , f )) := (hσf (h ), f f ) is called the semidirect product of H and F with respect to the homomorphism σ. 42 So the underlying set of a semidirect product is in any case the cartesian product, but the group multiplication is the componentwise multiplication only if σ : F −→ Aut(H) is the trivial homomorphism: σf = idH for all f ∈ F.

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Groups, Rings and Fields [Lecture notes] by Karl-Heinz Fieseler


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