38 Chapter 1. Groups I ∗ 1.40. (i) Give an example of two subgroups H and K of a group G whose union H ∪ K is not a subgroup of G. Hint. Let G be the four-group V. (ii) Prove that the union H ∪ K of two subgroups is itself a subgroup if and only if H is a subset of K or K is a subset of H. ∗ 1.41. Let G be a finite group with subgroups H and K. If H ⊆ K ⊆ G, prove that [G : H] = [G : K][K : H]. 1.42. If H and K are subgroups of a group G and |H| and |K| are relatively prime, prove that H ∩ K = {1}. Hint. If x ∈ H ∩ K, then x|H| = 1 = x|K|. ∗ 1.43. Let G be a group of order 4. Prove that either G is cyclic or x2 = 1 for every x ∈ G. Conclude, using Exercise 1.35 on page 27, that G must be abelian. ∗ 1.44. If H is a subgroup of a group G, prove that the number of left cosets of H in G is equal to the number of right cosets of H in G. Hint. The function ϕ: aH → Ha−1 is a bijection from the family of all left cosets of H to the family of all right cosets of H. 1.45. If p is an odd prime and a1,.. . , ap−1 is a permutation of {1, 2,.. . , p − 1}, prove that there exist i = j with iai ≡ jaj mod p. Hint. Use Wilson’s Theorem. Section 1.5. Homomorphisms An important problem is determining whether two given groups G and H are some- how the same. For example, we have investigated S3, the group of all permutations of {1, 2, 3}. The group SY of all the permutations of Y = {a, b, c} is a group dif- ferent from S3, because permutations of {1, 2, 3} are not permutations of {a, b, c}. But even though S3 and SY are different, they surely bear a strong resemblance to each other (see Example 1.57). More interesting is the strong resemblance of S3 to D6, the symmetries of an equilateral triangle. The notions of homomorphism and isomorphism will allow us to compare different groups. Definition. Let (G, ∗) and (H, ◦) be groups (we have displayed the binary opera- tions on each). A homomorphism23 is a function satisfying f(x ∗ y) = f(x) ◦ f(y) for all x, y ∈ G. If f is also a bijection, then f is called an isomorphism. Two groups G and H are called isomorphic, denoted by G ∼ = H, if there exists an isomorphism f : G → H between them. 23The word homomorphism comes from the Greek homo meaning “same” and morph mean- ing “shape” or “form.” Thus, a homomorphism carries a group to another group (its image) of similar form. The word isomorphism involves the Greek iso meaning “equal,” and isomorphic groups have identical form.

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