Preface
CHAPTER 0: SETS, INTEGERS, FUNCTIONS
1. CHAPTER 1
1.1. If A is a finite set having n elements, prove that A has exactly n 2 distinct subsets
1.2. For the given set and relations below, determine which define equivalence relations
1.3. Let a and b be two integers. If a|b and b|a, then show that a = ±b
1.4. Let p1, p2, · · ·, pn be distinct positive primes. Show that (p1 p2 · · · pn) + 1 is divisible by none of these primes
1.5. Prove that there are infinitely many primes
2. CHAPTER 2
2.1. If there are integers a, b, s, and t such that, the sum at+bs = 1, show that gcd(a, b) = 1
2.2. Show that if a and b are positive integers, then ab = lcm(a, b) · gcd(a, b)
2.3. Let S be any set. Prove that the law of multiplication defined by ab = a is associative
2.4. Assume that the equation xyz = 1 holds in a group G. Does it follow that yzx = 1? That yxz = 1? Justify your answer
2.5. Let G be a nonempty set closed under an associative product, which in addition satisfies: (a) There exists an e ∈ G such that ae = a for all a ∈ G. Prove that G must be a group under this product
2.6. If G is a group of even order, prove that it has an element a ≠ e satisfying a² = e
2.7. If G is a finite group, show that there exists a positive integer m such that a^m = e for all a ∈ G
2.8. If G is a group in which (ab)^i = a^i b^i for three consecutive integers i for all a, b ∈ G, show that G is abelian
2.9. If G is a group such that (ab)^2 = a^2 b^2 for all a, b ∈ G, then show that G must be abelian
2.10. Let a, b be elements of a group G. Assume that a has order 5 and a^3 b = b a^3. Prove that ab = ba
2.11. Let a and b be integers. (a) Prove that the subset aZ + bZ = {ak + bl | l, k ∈ Z} is a subgroup of Z. (b) Prove that a and b + 7a generate the subgroup aZ + bZ
2.12. Let H be the subgroup generated by two elements a, b of a group G. Prove that if ab = ba, then H is an abelian group
2.13. (a) Assume that an element x of a group has order rs. Find the order of x^r. (b) Let k be the order of x^r. Prove that in any group the orders of ab and of ba are equal
2.14. Let G be a group such that the intersection of all its subgroups which are different from {e} is a subgroup different from identity. Prove that every element in G has finite order
2.15. Show that if every element of the group G is its own inverse, then G is abelian
2.16. Let G be the set of all 2×2 matrices where a, b, c, d are integers modulo 2, such that ad − bc ≠ 0. Using matrix multiplications as the operation in G prove that G is a group of order 6
2.17. Let G be the group of all non-zero complex numbers a + bi (a, b real, but not both zero) under multiplication, and let H = { a + bi ∈ G | a² + b² = 1 }. Verify that H is a subgroup of G
2.18. Let G be a finite group whose order is not divisible by 3. Suppose that (ab)^3 = a^3 b^3 for all a, b ∈ G. Prove that G must be abelian
2.19. Let G be the group of all 2×2 matrices where a, b, c, d are integers modulo 3 and ad − bc ≠ 0. (b) If we modify the example of G in part (a) by insisting that ad − bc = 1, then what is |G|?
2.20. If H is a subgroup of G, and a ∈ G let aHa⁻¹ = { aha⁻¹ | h ∈ H }. Show that aHa⁻¹ is a subgroup of G
2.21. The center Z of a group G is defined by Z = { z ∈ G | zx = xz for all x ∈ G }. Prove that Z is a subgroup of G
2.22. If H is a subgroup of G, then by the centralizer CG(H) of H we mean the set { x ∈ G | xh = hx for all h ∈ H }. Prove that CG(H) is a subgroup of G
2.23. If a ∈ G define CG(a) = { x ∈ G | xa = ax }. Show that CG(a) is a subgroup of G. The group CG(a) is called the centralizer of a in G
2.24. If N is a normal subgroup of G and H is any subgroup of G prove that NH is a subgroup of G
2.25. Suppose that H is a subgroup of G such that whenever Ha ≠ Hb, then aH ≠ bH. Prove that gHg⁻¹ ⊆ H
2.26. Suppose that N and M are two normal subgroups of G and that N ∩ M = {e}. Show that for any n ∈ N, m ∈ M, nm = mn
2.27. If G is a group and H is a subgroup of index 2 in G, then prove that H is a normal subgroup of G
2.28. Show that the intersection of two normal subgroups of G is a normal subgroup of G
2.29. If N and M are normal subgroups of G, prove that NM is also a normal subgroup of G
2.30. If a cyclic group T of G is normal in G, then show that every subgroup of T is a normal subgroup in G
2.31. If N is a normal subgroup in the finite group such that (|G : N|, |N|) = 1. Show that any element x ∈ G satisfying x^{|N|} = e must be in N
2.32. Let G be a group in which for some integer n > 1, (ab)^n = a^n b^n for all a, b ∈ G. Show that (a) G^n = { x^n | x ∈ G } is a normal subgroup of G. (b) G^{n-1} = { x^{n-1} | x ∈ G } is a normal subgroup of G
2.33. Let P and Q be two normal p-subgroups of a finite group G. Show that PQ is a normal p-subgroup of G