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COURSE CODE: MTS 311
COURSE TITLE - Groups and Rings NUMBER OF UNITS: Three
Lecturer in Charge: Olusola J. Adeniran, Ph.D. E-mail: [email protected] Office: Room B213 COLNAS
COURSE CONTENT:
Groups, Examples of groups, Some elementary properties of groups, Subgroups, Cyclic groups, Cosets and Langrange’s theorem, Normal subgroups and Quotient groups, Homomorphisms, The Isomorphism Theorems, Group Actions, orbits and stabilizers, Conjugacy, Class equation of a finite group, Cauchy’s Theorem, The structure of p-groups, The Sylow’s theorems, Some applications of Sylow’s theorems, Simple groups, Solvable groups
COURSE REQUIREMENTS
- COURSE STATUS: Compulsory
- PREREQUISITE: MTS 211 - Abstract Algebra
RECOMMENDED TEXTS
- Elementary Abstract and Linear Algebra by Ilori & Akinyele, University of Ibadan Press
- Abstract Algebra by Aderemi Kuku, University of Ibadan Press
- A first course in Abstract Algebra by J.B. Fraleigh
- A course in Algebra by E.B. Vinberg, American Mathematical Society, 2001
1 Groups
A binary operation ⋆ on a set G associates to elements x and y of G a third element x ⋆ y of G. For example addition and multiplication are binary operations of the set of all integers.
Definition 1.1 A group G consists of a set G together with a binary operation ⋆ for which the following properties are satisfied:
- (x ⋆ y) ⋆ z = x ⋆ (y ⋆ z) for all x, y&z of G (the associative law)
- there exist an element e of G (known as the identity element of G) such that e ⋆ x = x = x ⋆ e, for all element x of G.
- for each element x of G there exists an element x′^ (known as the inverse of x) such that x⋆x = e = x′^ ⋆x (where e is the identity element of G).
1.0.1 Examples of Groups
- The set of integers, rational numbers, real numbers and complex numbers are Abelian groups together with the binary operation of addition.
- The set of non-zero rational numbers, non-zero real numbers and non-zero complex numbers are are also Abelian groups with the binary operation of multiplication
- For each positive integer m ZZm of congruency classes of integers modulo m is a group, where the group operation is addition of congruence classes.
- For each positive integer n the set of all singular n × n matrices is a group where the group operation is matrix multiplication. These groups are not Abelian for n ≥ 2.
1.1 Some elementary properties of groups
In the following the some properties of a group G using multiplicative notation and denoting the identity element e are given.
Lemma 1.1 A group G has exactly one identity element e such that xe = ex = e for all x ∈ G
Proof Suppose that f is an element of G with the property that f x = x foe all elements x of G. Then in particular f = f e = e. Similarly one can show that e is the only element of G satisfying xe = x for all element x of G.
Lemma 1.2 Every element x of G has exactly one inverse x−^1
Proof From the axioms of a group, G contains at least one element x−^1 which satisfies xx−^1 = e and x−^1 x = e. If z is any element of G which satisfies xz = e then z = ez = (x−^1 x)z = x−^1 (xz) = x−^1 e = x−^1. Similarly if w is any element of G which satisfies wx = e then w = x−^1. In particular we conclude that the inverse x−^1 of x is uniquely determined. This ends the proof.
Lemma 1.3 Let x and y be elements of a group G. Then (xy)−^1 = y−^1 x−^1
From the axioms of a group (xy)(y−^1 x−^1 ) = x(y(y−^1 x−^1 )) = x((yy−^1 )x−^1 ) = x(ex−^1 ) = xx−^1 = e. Similarly (y−^1 x−^1 )(xy) = e, and thus y−^1 x−^1 is the inverse of xx−^1 as required.
NOTE In particular that (x−^1 )−^1 = x for all elements x of a group G, since x has the properties that characterize the inverse of the inverse x−^1 of x.
1.3.1 Examples of Cyclic groups
- The group ZZ of integers under addition is a cyclic group generated by 1.
- Let n be a positive integer. The set ZZn of congruence classes of integers modulo n is a cyclic group of order n withe respect to the operation of addition.
- The group of all rotations of the plane about the origin through an integer multiple of 2π/n radians is a cyclic group of order n. This group is generated by an anticlockwise rotation through an angle of 2 π/n radian.
1.4 Cosets and Lagranges Theorem
Definition 1.5 Let H be a subgroup of a group G. A left coset of H in G is a subset of G that is of the form xH, where x ∈ G and xH = {y ∈ G : y = xh for some h ∈ H}
Similarly, a right coset of H in G is a subset of G that is of the form Hx, where x ∈ G and
Hx = {y ∈ G : y = hx for some h ∈ H}.
NOTE that a subgroup H of a group G is itself a left coset of H in G.
Lemma 1.6 Let H be a subgroup of a group G. Then the left coset of H in G have the following properties:
- x ∈ xH for all x ∈ B
- If x and y are elements of G, and if y = xa for some a ∈ H, then xH = yH
- If x and y are elements of G, and if xH ∩ yH is non-empty then xH = yH.
Proof Let x ∈ G. Then x = xe, where e is the identity element of G. But e ∈ H. It follows that x ∈ xH hence 1 is proved. Let x and y be elements of G where y = xa for some a ∈ H. Then yh = x(ah) and xh = y(a−^1 h) for all h ∈ H. Moreover, ah ∈ H and a−^1 ∈ H for all h ∈ H, since H is a subgroup of G. It follows that yH ⊂ xH and xH ⊂ yH and 2 is proved. Finally, suppose that xH ∩ yH is non-empty for some elements x and y of G. Let z be an element of xH ∩ yH. Then z = xa for some a ∈ H, and z = yb for some b ∈ H. It follows from 2 that zH = xH and zH = yH. Therefore xH = yH. This proves 3 .
Lemma 1.7 Let H be a finite subgroup of a group G. Then each left coset of H in G has the same number of elements as H.
Proof To be provided during Lecture
Theorem 1.2 (Lagrange’s theorem) Let G be a finite group, and let H be a subgroup of G. Then the order of H divides the order of G.
Proof Each element of G belongs to at least one left coset of H in G and no element of can belong to two distinct left cosets of H in G (see Lemma 2.6). Therefore every element of G belongs to exactly one left coset of H. Moreover, each left coset of H contains |H| elements (Lemma 2.7). Therefore,|G| = n|H| where n is the number of left cosets of H in G. Hence the result follows.
Definition 1.6 Let H be a subgroup of a group G. If the number of left cosets of h in G is finite then the number of such cosets is referred to as the index of H in G, denoted by [H : G].
The proof of Lagrange’s Theorem shows that the index [G : H] of a subgroup H of a finite group G given by [G : H] = |G|/|H|.
Corollary 1.1 Let x be an element of a finite group G. Then the order of x divides the order of G.
Proof To be provided during Lecture
Corollary 1.2 Any finite group of prime order is cyclic.
Proof To be provided during Lecture
1.5 Normal subgroups and quotient groups
Let A and B be subsets of a group G. The product AB of the sets A and B is defined by
AB = {xy : x ∈ Aandy ∈ B}
We denote {x}A and A{x} for all x ∈ G and subsets A ⊆ G. The Associative Law for multiplication of elements of G ensures that (AB)C = A(BC) for all subsets A, B and C of G. We can therefore use the notation ABC to denote (AB)C and A(BC); and we can use analogous notation to denote the product of four or more subsets of G. If A, B and C are subsets of a group G, and if A ⊂ B then clearly AC ⊂ BC and CA ⊂ CB. Note that if H is a subgroup of the group G and if x is an elements of G then xH is the left coset of H in G that contains the element x. Similarly Hx is the right coset of H in G that contains the element x. If H is a subgroup of G then HH = H. Indeed, HH ⊂ H, since the product of two elements of a subgroup H is itself an element of H. Also, H ⊂ HH since h = eh for any element h ∈ H, where e, the identity element of G belongs to H.
Definition 1.7 A subgroup N of a group G is said to be a normal subgroup if G if xnx−^1 ∈ N for all n ∈ N and x ∈ G.
The notation ‘N ▹ G’ signifies ‘N is a normal subgroup of G’.
Definition 1.8 A non-trivial group G is said to be simple if the only normal subgroups of G are the whole of G and the trivial subgroup {e} whose only element is the identity element of e of G.
Lemma 1.8 Every subgroup of an Abelian group is a nornmal subgroup
Proof To be provided during Lecture
EXAMPLE Let S 3 be the group of permutations of the set { 1 , 2 , 3 }and let H be the subgroup of S 3 consist- ing of the identity permutation and the transposition (12). Then H is not normal in G since (23)−^1 (12)(23) = (23)(12)(23) = (13) and (13) does not belong to the subgroup H.
Proposition 1.1 A subgroup N of a normal subgroup of G¿ Let x be an element of G. Then xN x−^1 = N for all element x ∈ G
Proof To be provided during Lecture
Corollary 1.3 A sugroup N of a group G is a normal subgroup of G if and only if xN = N x for all element x of G.
Definition 1.11 An isomorphism θ : G −→ K between group G and K is a homomorphism that is also a bijective mapping G onto K. Two groups G and K are said to be isomorphic if there exists an isomorphism mapping G onto K.
EXAMPLE Let D 6 be the group of symmetries of an equilateral triangle in the plane with vertices X, Y and Z and let S 3 be the group of permutations of the set {X, Y, Z}. The function which sends a symmetry of the triangle to the corresponding permutation of its vertices is an isomorphism between the dihedral group D 6 of order 6 and the symmetric group S 3
EXAMPLE
Let R be the group of real numbers with the operation of addition and let R+^ be the group of strictly positive real numbers with the operation of multiplication. The function exp : R −→ R+^ that sends each real number x to the positive real number ex^ is an isomorphism: it is both homomorphism of groups and a bijection. The inverse of this isomorphism is the function log : R+^ −→ R that sends each strictly positive real number to its natural logarithm
Definition 1.12 The following are some terminologies regarding homomorphism:
- A monomorphism is an injective homomorphism.
- An epimorphism is a surjective homomorphism.
- An endomorphism is a homomorphism mapping a group into itself.
- An automorphism is an isomorphism mapping a group onto itself.
Definition 1.13 The kernel Kerθ of the homomorphism θ : G −→ K is the set of all elements of G that are mapped by θ onto the identity element of K.
EXAMPLE Let the group operation on the set {+1, − 1 } be multiplication, and let θ : ZZ −→ {+1, − 1 } be the homo- morphism that sends each integer n to (−1)n. Then the kernel of the homomorphism θ is the subgroup of ZZ consisting of all even numbers.
Lemma 1.11 Let G and K be groups, and let θ : G −→ K be a homomorphism from G to K. Then the kernel kerθ of θ is a normal subgroup of G.
Proof To be provided during Lecture
NOTE If N is a normal subgroup of some group G then N is the kernel of the quotient homomorphism θ : G −→ G/N that sends g ∈ G to the coset gN. It follows therefore that a subset of a group G is a normal subgroup of G if and only it it is the kernel of some homomorphism.
Proposition 1.3 Let G and K be groups, let θ : G −→ K be a homomorphism from G to K, and let N be a normal subgroup of G. Suppose that N ⊂ kerθ. Then the homomorphism θ : G −→ K induces a homomorphism θ : G/N −→ K sending gN ∈ G/N to θ(g). Moreover
Proof To be provided during Lecture
Corollary 1.4 Let G and K be groups, and let θ : G −→ K be a homomorphism. Then θ(G) ∼= G/kerθ.
Proof To be provided during Lecture
1.7 The Isomorphism Theorems
Lemma 1.12 Let G be a group, let H a subgroup of G, and let N be a normal subgroup of G. Then the set HN is a subgroup of G, where HN = {hn : handn ∈ N }.
Proof To be provided during Lecture
Theorem 1.3 (First Isomorphism Theorem) Let G be a group, and let H be a subgroup of G, and let N be a normal subgroup of G. Then
HN N
∼= H
N ∩ H
Proof To be provided during Lecture
Theorem 1.4 (Second Isomorphism Theorem) Let M and N be normal subgroups of a group G, where M ⊂ N. Then
G N
∼= G/M
N/M
Proof To be provided during Lecture
1.8 Group Actions, Orbits and Stabilizers
Definition 1.14 A left action of a group G on a set X associates to each g ∈ G and x ∈ X an element g · x of X in such a way that g · (h · x) = (gh) · x and 1 · x = x for all g.h ∈ G and x ∈ X, and 1 denotes the identity element of G Given a left action of a group G on a set X, the orbit of an element x of X is the subset {g · x : a ∈ G} of X and the stabilizer of x is the subgroup {g ∈ G : g · x = x} of G
Lemma 1.13 Let G be a finite group which acts on a set X on the left. Then the orbit of an element x of X contains [G : H] elements, where [G : H] is the index of stabilizer H of x in G.
Proof To be provided during Lecture
1.9 Conjugacy
Definition 1.15 Two elements h and k of a group G are said to be conjugate if k = hhg−^1 for some g ∈ G
NOTE
- It can readily be verified that the relation of conjugacy is reflexive, symmetric and transitive and therefore an equivalence relation on a group G.
