Abstract Algebra Theory and Applications - Computer Science ...

EXERCISES 201(c) H ∗ (H ∩ K)/H ∗ (H ∩ K ∗ ) ∼ = K ∗ (H ∩ K)/K ∗ (H ∗ ∩ K)∼= (H ∩ K)/(H ∗ ∩ K)(H ∩ K ∗ ).[Hint: Use the diagram in Figure 11.1. The Zassenhaus Lemma is oftenreferred to as the Butterfly Lemma because of this diagram.]H ∗HK✂◗◗❇✂ ◗❇✂◗◗❇✂H ∗ (H ∩ K) ◗ H ∩ K K ∗ (H ∩ K)◗✑ ✑✑✑✑✑✑✑ ❇✂◗❇✂❇✂H ∗ (H ∩ K ∗ ) K ∗ (H ∗ ❇✂∩ K)✟✁◗◗◗◗✑ ✑✑✑❆❍ ❇✂ ✟❍❍❍❍❍❍❍✟❇✂ ✟✟ ✁❆❇✂ ✟✟ ✁ ✑◗ ❆ ❇❇✟✂✁ ✑ ◗◗◗◗◗◗◗ ❆❅ ✁ ✑✑❆❅ ✁ ✑✑ (H ∗ ∩ K)(H ∩ K ∗ ❆ ❅ ✁)✑❆❆ ❅✑✁H ∗ ∩ KH ∩ K ∗K ∗Figure 11.1. The Zassenhaus Lemma23. Schreier’s Theorem. Use the Zassenhaus Lemma to prove that two subnormal(normal) series of a group G have isomorphic refinements.24. Use Schreier’s Theorem to prove the Jordan-Hölder Theorem.Programming ExercisesWrite a program that will compute all possible abelian groups of order n. What isthe largest n for which your program will work?References and Suggested ReadingsEach of the following references contains a proof of the Fundamental Theorem ofFinitely Generated Abelian Groups.[1] Hungerford, T. W. Algebra. Springer-Verlag, New York, 1974.