Group Theory
Definition 1. Let G be a set. A binary operation on G is a function that assigns each ordered pair of elements of G an element of G.
Definition 2. Let G be a set together with a binary operation (usually called multiplication) that assigns to each ordered pair (a, b) of elements of G an element in G denoted by ab. We say G is a group under this operation if the following three properties are satisfied.
- i. Associativity. The operation is associative; that is, (ab)c = a(bc) for all a, b, c in G.
- ii. Identity. There is an element e (called the identity) in G such that ae = ea = a for all a in G.
- iii. Inverses. For each element a in G, there is an element b in G (called an inverse of a) such that ab = ba = e.
Theorem 1. In a group G, there is only one identity element
Theorem 2. In a group G, the right and left cancellation laws hold; that is, ba = ca implies b = c, and ab = ac implies b = c.
Theorem 3. For each element a in a group G, there is a unique element b in G such that ab = ba = e.
Theorem 4. For group elements a and b, (ab)–1 = a–1b–1.
Group Theory Problems
1. Give examples to verify that subtraction is not associative.
2. Determine if the following sets G with the operation indicated form a group. If not, point out which of the group axioms fail.
- (a) G = set of all integers, a * b = a – b.
- (b) G = set of all integers, a * b = a + b + ab.
- (c) G = set of nonnegative integers, a * b = a + b.
- (d) G = set of all rational numbers ≠ –1, a * b = a + b + ab.
- (e) G = set of all rational numbers with denominator divisible by 5 (written so that numerator and denominator are relatively prime), a * b = a + b.
- (f) G a set having more than one element, a * b = a for all a, b ∈ G.
3. Give two reasons why the set of odd integers under addition is not a group.
4. Prove that the set of all 2 × 2 matrices with entries from ℝ and determinant +1 is a group under matrix multiplication.
5. Show that {1, 2, 3} under multiplication modulo 4 is not a group but that {1, 2, 3, 4} under multiplication modulo 5 is a group.
6. An abstract algebra teacher intended to give a typist a list of nine integers that form a group under multiplication modulo 91. Instead, one of the nine integers was inadvertently left out, so that the list appeared as 1, 9, 16, 22, 53, 74, 79, 81. Which integer was left out?
7. Give an example of a group with 105 elements. Give two examples of groups with 44 elements.
8. Give an example of group elements a and b with the property that a–1ba ≠ b.
9. List the members of H = {x2 | x ∈ D4} and K = {x ∈ D4 | x2 = e}.
10. (Law of exponents for Abelian groups) Let a and b be elements of an Abelian group and let n be any integer. Show that (ab)n = anbn. Is this also true for non-Abelian groups?
11. Prove that a group G is Abelian if and only if (ab)–1 = a–1b–1 for all a and b in G.
12. The integers 5 and 15 are among a collection of 12 integers that form a group under multiplication modulo 56. List all 12.
13. Prove that every group table is a Latin square (that is, each element of the group appears exactly once in each row and each column.
14. Suppose the table below is a group table. Fill in the blank entries.
15. Prove that if (ab)2 = a2b2 in a group G, then ab = ba.
16. Prove that the set of all rational numbers of the form 3m6n, where m and n are integers, is a group under multiplication.
17. Let a, b, and c be elements of a group. Solve the equation axb = c for x. Solve a–1xb = c for x.
18. Let G be a group with the following property: Whenever a, b, and c belong to G and ab = ca, then b = c. Prove that G is Abelian (cross cancellation implies commutativity).
19. Give an example of a group with elements a, b, c, d, and x such that axb = cxd but ab ≠ cd (hence middle cancellation is not valid in groups).
20. Suppose that G is a group with the property that for every choice of elements in G, axb = cxd implies ab = cd. Prove that G is Abelian (middle cancellation implies commutativity).
21. Let G be a set with an operation * such that:
i. G is closed under *.
ii. * is associative.
iii. There exists an element e ∈ G such that e * x = x for all x ∈ G.
iv. Given x ∈ G, there exists a y ∈ G such that y * x = e.
Prove that G is a group (thus you must show that x * e = x and x * y = e for e, y as above).
22. Let G be a finite nonempty set with an operation * such that:
i. G is closed under *.
ii. * is associative.
iii. Given a, b, c ∈ G with a * b = a * c, then b = c.
iv. Given a, b, c ∈ G with b * a = c * a, then b = c.
Prove that G must be a group under *. Give an example to show that the result can be false if G
is an infinite set.
23. If G is a group in which (ab)i = aibi for three consecutive integers i, prove that G is abelian. Show that the result of would not always be true if the word three were replaced by two.
24. Let G be a group in which (ab)3 = a3b3 and (ab)5 = a5b5 for all a, b ∈ G. Show that G is abelian.
25. Let R be any rotation in some dihedral group and F any reflection in the same group. Prove that RFR = F.
26. If A, B are subgroups of G, show that A ∩ B is a subgroup of G.
27. What is the cyclic subgroup of ℤ generated by –1 under +?
28. Let S3 be the symmetric group of degree 3. Find all the subgroups of S3.
29. Verify that Z(G), the centre of G, is a subgroup of G.
30. Prove that a cyclic group is abelian.
31. If G is cyclic, show that every subgroup of G is cyclic.
32. If G has no proper subgroups, prove that G is cyclic.
33. If G has no proper subgroups, prove that G is cyclic of order p, where p is a prime number.
34. If A, B are subgroups of an abelian group G, let AB = {ab | a ∈ A, b ∈ B}. Prove that AB is a subgroup of G.
35. Give an example of a group G and two subgroups A, B of G such that AB is not a subgroup of G.
36. If A, B are subgroups of G such that b–1Ab ⊂ A for all b ∈ B, show that AB is a subgroup of G.
37. If A and B are finite subgroups, of orders m and n, respectively, of the abelian group G, prove that AB is a subgroup of order mn if m and n are relatively prime.
38. Show that any group of order 4 or less is abelian.
39. Show that a group of order 5 must be abelian.
40. If G is a group in which a2 = e for all a ∈ G, show that G is abelian.
41. If G is a finite group of even order, show that there must be an element a ≠ e such that a = a–1.
42. If G is a finite group, prove that, given a ∈ G, there is a positive integer n, depending on a, such that an = e. Also show that there is an integer m > 0 such that am = e for all a ∈ G.
43. Let G be a finite group. Show that the number of elements x of G such that x3 = e is odd. Show that the number of elements x of G such that x2 ≠ e is even.
44. Let S be a set having an operation * which assigns an element a * b of S for any a, b ∈ S. Let us assume that the following two rules hold:
i. If a, b are any objects in S, then a * b = a.
ii. If a, b are any objects in S, then a * b = b * a.
Show that S can have at most one object.
45. Show that a set having n elements has 2n subsets. (b) If 0 < m < n, how many subsets are there that have exactly m elements?