Exercise 3.6.12 Prove the First Isomorphism Theorem of Rings: If o: R→S is a surjective ring homomorphism, then there exists a unique ring isomorphism

Exercise 3.6.11 Suppose R is a ring and JCR is a (two-sided) ideal (see Exercise 3.3.13 for the definition of an ideal). The quotient additive group R/J is the set of cosets, which in the additive notation have the form r +J, for r ER.

Exercise 6. INVERSE TRANSFORM Determine the signal x(t) by computing the inverse Fourier transform of the following X(jw).

Exercise 3.6.4 Suppose that G₁, G₂ are groups, N₁ ◄G₁, N₂ ◄ G₂ are normal subgroups. Prove that N₁ × N₂ ◄ G₁ x G₂, and (G₁ × G₂)/(N₁ × N₂) ≈G₁/N₁ × G₂/N₂. Hint: Theorem 3.6.5.

Exercise 3.7.2 Suppose G is a finite group, H, KG are normal subgroups, ged(|H|, |K|) = 1, and |G| = |H||K|. Prove that G H x K.

Exercise3.7.6 Prove that for all n≥2，we have Sn≌An为Z2,where Sn is the symmetric group and Anthe alternating subgroup (consisting of the even permutations).Hint:use Proposition 3.7.7.You have to find asubgroup K Sn with KZ2 (c.f.Erample 3.7.8).

Exercise 4.1.1 Prove Proposition 4.1.5: Suppose Gx X → X is an action. Then for all z € X, stabĠ(z) is a subgroup of G and the kernel K of the action is the intersection of all stabilizers