We can say that "o" is the binary operation on set G if: G is a non-empty set & G * G = { (a,b): a, b∈ G } and o: G * G --> G. Here, aob denotes the image of ordered pair (a,b) under the function/operation o.
Example - "+" is called a binary operation on G (any non-empty set ) if & only if: a+b ∈G; ∀ a,b ∈G and a+b give the same result every time when added.
Algebraic Structure:
A non-empty set G equipped with 1/more binary operations is called an algebraic structure.
Example - a. (N,+) and b. (R, + , .), where N is a set of natural numbers & R is a set of real numbers. Here ' . ' (dot) specifies a multiplication operation.
Group
An algebraic structure (G, o) where G is a non-empty set & 'o' is a binary operation defined on G is called a Group if the binary operation "o" satisfies the following properties -
Closure - a ∈ G ,b ∈ G => aob ∈ G ; ∀ a,b ∈ G
- Associativity - (aob)oc = ao(boc) ; ∀ a,b,c ∈ G.
- Identity Element - There exists e in G such that aoe = eoa = a; ∀ a ∈ G (Example - For addition, identity is 0).
- Existence of Inverse - For each element a ∈ G; there exists an inverse(a-1)∈ G such that: aoa-1 = a-1oa = e.
Abelian Group
An algebraic structure (G, o) where G is a non-empty set & 'o' is a binary operation defined on G is called an abelian Group if it is a group (i.e. , it satisfies G1, G2, G3 & G4) and additionally satisfies
Commutative - aob = boa ∀ a,b ∈ GNormal Subgroup
Let G be an abelian group & the composition in G has been denoted by multiplicity.
Let H be any subgroup of G. If x is an arbitrary element of G, the Hx is the right coset of H in G & xH is the left coset of H in G, then G is called a normal subgroup if -
Hx = xH ; ∀x ∈ G or
xhx-1 ∈ H ; ∀x ∈ G & h ∈ H
Quotient Group
Let G be any group & let N be any normal Subgroup of G. If 'a' is an element of G, then aN is a left coset of N in G. Since N is normal in G, aN = Na ( left coset = right coset).
We can say that Na is the coset of N in G.
G/N denotes the set of all the cosets of N in G.
Quotient/Factor Group = G/N = {Na ; a ∈ G } = {aN ; a ∈ G} (As aN = Na)If G is a group & N is a normal subgroup of G, then, the sets G/N of all the cosets of N in G is a group with respect to multiplication of cosets in G/N. It is called the quotient / factor group of G by N.
Sometimes it is called 'Residue class of G modulo N'.
If the composition in the group is addition, '+', then G/H is defined as:
Quotient/Factor Group = G/N = {N+a ; a ∈ G } = {a+N ; a ∈ G} (As a+N = N+a)NOTE - The identity element of G/N is N.
Example 1 - Consider the group G with addition modulo 6 where G = {0, 1, 2, 3, 4, 5}. Let N = {0, 3),
then the quotient/ factor group is :
G/N = { aN ; a ∈ G } = { a{0,3} ; a ∈ {0, 1, 2, 3, 4, 5}}
= {0{0,3}, 1{0,3}, 2{0,3}, 3{0,3}, 4{0,3}, 5{0,3} }
= { {(0+0) mod6 , (0+3) mod6 }, { (1+0) mod6 , (1+3) mod6 } , { (2+0) mod6 , (2+3) mod6 } , { (3+0) mod6 , (3+3) mod6 }, { (4+0) mod6 , (4+3) mod6 }, { (5+0) mod6 , (5+3) mod6 } }
= {{0,3}, {1,4}, {2,5}, {3,0}, {4,1}, {5,2} }
= {{0,3}, {1,4}, {2,5}}
Example 2 - Let G = {1, -1, i, -i } and H = {1, -1}; H is the normal subgroup of G in binary operation ',' . What will be the quotient group; G/H?
G/N = { aN ; a ∈ G } = {a{1,-1} ; a ∈ {1,-1,i,-i}
= {1.{1,-1}, -1.{1,-1}, i{1,-1}, -i.{1,-1}}
={{1.1,1.-1}, {-1.1,-1.-1}, {i.1, i.-1}, {-i.1, -i.-1}}
={{1,-1}, {-1,1}, {i,-i}, {-i,i}}
={ {1,-1}, {i,-i}}
In other words, we can say that if G is a group & N is a normal subgroup of G, then G/N of all the cosets of N in G together with a binary composition defined by:
NaNb = Nab ; where Na ∈ G/N, Nb ∈ G/N is a group.G/N is called the quotient group of G by N.
Properties of Quotient/ Factor group
- If N is a normal subgroup of a finite group G, then -
O(G/N) = O(G)/O(N), where : O(G/N) => No of distinct right / left cosets of N in G. - If N is a normal subgroup of a finite group G such that the index of N in G is prime, the factor group G/N is cyclic.
- The factor group of an abelian group is abelian, but the converse is not true.
- Every factor group of a cyclic group is cyclic but the converse is not true.
Solved Examples on Quotient Group in Group Theory
Problem 1:
Let G be the group of real numbers under addition, and let H be the subgroup of integers. Describe the elements of the quotient group G/H and its operation.
Solution:
The elements of G/H are cosets of the form a + H, where a is a real number. Each coset can be represented by a real number in the interval [0, 1), as any real number can be written as the sum of an integer and a number in [0, 1). The operation in G/H is addition modulo 1. Thus, G/H is isomorphic to the group of real numbers modulo 1, often denoted as R/Z.
Problem 2:
Let G be the group of invertible 2×2 matrices over real numbers, and H be the subgroup of matrices with determinant 1 (the special linear group SL(2,R)). What is the order of the quotient group G/H?
Solution:
The determinant function det: G → R* (where R* is the multiplicative group of non-zero real numbers) is a surjective homomorphism with kernel H. By the first isomorphism theorem, G/H ≅ R*. Since R* has infinite order, G/H also has infinite order.
Problem 3:
Let G be the symmetric group S4 and H be the alternating group A4. Describe the elements of G/H and determine its order.
Solution:
Since A4 is a normal subgroup of S4 of index 2, G/H has two cosets: A4 and S4 \ A4 (the set of odd permutations). The order of G/H is 2, making it isomorphic to Z2.
Problem 4:
Let G be the group of complex numbers under multiplication, and H be the subgroup of complex numbers with absolute value 1. Describe the quotient group G/H.
Solution:
Each coset in G/H can be represented by a positive real number (the absolute value of any complex number in the coset). The operation in G/H is multiplication of these positive real numbers. Thus, G/H is isomorphic to the multiplicative group of positive real numbers, (R+, ×).
Problem 5:
Let G be the group of 3×3 upper triangular matrices with real entries and non-zero diagonal elements under matrix multiplication. Let H be the subgroup of matrices in G with 1s on the diagonal. Determine the structure of G/H.
Solution:
The diagonal elements of matrices in G form a homomorphic image of G that is isomorphic to (R*)³. The kernel of this homomorphism is H. By the first isomorphism theorem, G/H ≅ (R*)³.
Problem 6:
Let G = Z × Z and H = {(2a, 2b) | a, b ∈ Z}. Determine the structure of G/H.
Solution:
G/H has four cosets: (0,0)+H, (1,0)+H, (0,1)+H, and (1,1)+H. The quotient group is isomorphic to Z2 × Z2, the Klein four-group.
Problem 7:
Let G be the group of quaternions and H be the subgroup {1, -1}. Describe the quotient group G/H.
Solution:
The quaternion group has 8 elements: {±1, ±i, ±j, ±k}. In G/H, each pair of antipodal elements forms a coset. Thus, G/H has 4 elements and is isomorphic to the Klein four-group Z2 × Z2.
Problem 8:
Let G be the group of invertible n×n matrices over a field F, and H be the subgroup of scalar matrices (multiples of the identity matrix). Describe the quotient group G/H.
Solution:
G/H is isomorphic to the projective general linear group PGL(n,F). It consists of equivalence classes of matrices where two matrices are equivalent if they differ by a scalar multiple.
Problem 9:
Let G be the additive group of polynomials with real coefficients, and H be the subgroup of polynomials divisible by (x² + 1). Describe the quotient group G/H.
Solution:
Every polynomial in G can be uniquely written as q(x)(x² + 1) + (ax + b) for some polynomial q(x) and real numbers a and b. Thus, G/H is isomorphic to the group of linear polynomials ax + b under addition, which is isomorphic to R².
Problem 10:
Let G be the group of orientation-preserving isometries of a regular tetrahedron, and H be the subgroup of rotations about a fixed axis through a vertex and the center of the opposite face. Determine the structure of G/H.
Solution:
G is isomorphic to A4, which has 12 elements. H is a cyclic subgroup of order 3. The quotient G/H has 4 cosets, each corresponding to one face of the tetrahedron. The quotient group G/H is isomorphic to Z4, representing the 4-fold rotational symmetry around the axis perpendicular to all rotation axes in H.
Practice Problems on Quotient Group in Group Theory
1. Let G be the group of integers under addition and H be the subgroup of even integers. Describe the elements of G/H and determine its order.
2. Let G be the symmetric group S3 and H be the subgroup {e, (1 2 3), (1 3 2)}. Is H normal in G? If so, describe the elements of G/H.
3. Let G be the group of non-zero real numbers under multiplication and H be the subgroup of positive real numbers. Describe the structure of G/H.
4. Let G be the group of 2x2 matrices with real entries and non-zero determinant under matrix multiplication. Let H be the subgroup of matrices with determinant 1. Describe the quotient group G/H.
5. Let G be the group of complex numbers under addition and H be the subgroup of integers. Describe the elements of G/H and its operation.
6. Let G be the group Z6 × Z4 and H be the subgroup {(0,0), (3,2)}. Determine the order of G/H and describe its elements.
7. Let G be the quaternion group and H be the subgroup {1, -1, i, -i}. Describe the quotient group G/H.
8. Let G be the group of rotations of a cube and H be the subgroup of rotations that fix a particular face. Determine the order of G/H and describe what its elements represent.
9. Let G be the additive group of polynomials with real coefficients of degree less than or equal to 2, and H be the subgroup of constant polynomials. Describe the structure of G/H.
10. Let G be the group GL(2,R) of invertible 2x2 matrices over real numbers, and H be the subgroup of upper triangular matrices in G. Determine whether H is normal in G, and if so, describe G/H.
Summary
Quotient groups are fundamental structures in group theory that allow us to study the relationship between a group and its subgroups. They are formed by partitioning a group G into cosets with respect to a normal subgroup H, resulting in a new group G/H. This concept is crucial for understanding group homomorphisms, isomorphism theorems, and the structure of groups in general. Quotient groups provide a way to simplify complex group structures by "factoring out" a subgroup, revealing important properties of the original group. They are widely used in various branches of mathematics, including abstract algebra, topology, and number theory, and have applications in physics and computer science.
