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A group is a collection of components or elements that have been put together to fulfill a task. We are all familiar with the concept of sets in set theory. When any two of its constituents are merged by a mathematical operation to generate the third element from the same set that fits the four assumptions of closure, associativity, invertibility, and identity, it is termed as Group theory axioms. We also have the Lagrange theorem in group theory, which is an important theorem.
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Key terms: group, set, properties, algebraic, associative, component
Group Theory
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Group theory is a branch of mathematics that analyses the algebraic structures known as groups. Other well-known algebraic structures, such as rings, fields, and vector spaces can also be regarded as groups with extra operations and axioms. Groups appear often in mathematics, and group theory approaches have affected many aspects of algebra. Symmetry groups may be used to describe a variety of physical systems, including crystals and the hydrogen atom, as well as three of the four fundamental forces in the universe. Public key cryptography relies on group theory as well.
Properties of Group Theory
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The "Group Operation" is the operation (or formula) by which a group is formed, and a set is considered as a group under that operation. Given the components A, B, C with binary operations between A and B, and then consider if the notation "AB" forms a group
- Conclusion: If A and B are two components in the G group, then "AB" is also in "G."
- Associativity: Since the defined multiplication is associative, if all A, B, and C belong to the same group, "G," then (AB) C = A. (BC).
- Identity: There is an identity element, I, for each component, A, such that IA= AI= A.
- Inverse: Each component should have an inverse; thus, the set includes a component B= A' for every component A under G, resulting in AA' = A'A = I.
- A group must have at least one component with a unique isomorphism, with the trivial group being the single-element group.
- If A and B are components of G, the group is abelian, and AB = BA.
- A monoid is defined as a collection of invertible components of a group.
The addition of two integers yields an integer, which is the most typical example that meets these axioms. As a result, the closure property is met. In addition, the associative condition is satisfied by adding integers. In the group, there is an identity element named zero, which when added to any integer returns the original number. Also, for any integer, there is an inverse such that when they are combined, the result is zero. In the case of the addition operation of two integers, all of the group axioms are met.
Axioms and Proofs
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Axiom 1: If G is a group that has a and b as its elements, such that a, b ∈ G, then (a × b)-1 = a-1 × b-1
Proof:
To prove: (a × b) × a-1 × b-1 = I, where I is the identity element of G.
L.H.S = (a × b) × a-1 × b-1
=> a × (b × b-1) × a-1
=> a × I × a-1(by associative axiom)
=> (a × I) × a-1(by identity axiom)
= a × a-1 (by identity axiom)
= I (by identity axiom)
= R.H.S
Hence, proved.
Axiom 2: If in a group G, ‘x’, ‘y’ and ‘z’ are three elements such that x × y = z × y, then x = z.
Proof: Let us assume that x × y = z × y. (i)
As 'y' is a member of group G, it follows that there must be some 'a' in G with identity element I, as follows:
y × a = I (ii)
On multiplying both sides of (i) by ‘a’ we get,
x × y × a = z × y × a
x × (y × a) = z × (y × a) (by associativity)
From eq.(ii);
a × I = c × I [using (ii)]
a = c (by identity axiom)
This is often referred to as the cancellation law.
Hence, proved.
Application of Group Theory
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- Although group theory is the study of symmetry, group theory may be used to investigate any object or system attribute that is invariant under change.
- The Rubik's cube solving algorithm is based on group theory.
- The Lorentz group is a physics concept that expresses the basic symmetry of many fundamental laws of nature. In physics, chemistry, and material science, symmetry is used in many fundamental rules of nature.
- Group theory is also used in Galois theory, harmonic analysis, combinatorics, Algebraic topology, Algebraic geometry, Algebraic number theory and Cryptography.
Main classes of Groups
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The number of types of groups studied has increased gradually, from finite permutation groups and well-known instances of matrix groups to abstract groups that may be specified using generators and relations. As a result, we may classify groups into the following categories.
Permutation groups
Permutation groups are mathematical groups whose members are permutations of a given set M and whose group operation is the composition of permutations in G.
Matrix groups
In mathematics, a matrix group is a set G of invertible matrices with the operation of matrix multiplication over a defined field K. A linear group is an isomorphic group to a matrix group
Transformation groups
The term "transformation group" refers to a subgroup of an automorphism group. A transformation group is similar to a symmetry group in that it often consists of all transformations that retain a specific structure.
Abstract groups
A presentation of generators and relations is a common approach to defining an abstract group. The production of a factor group, or quotient group, G/H, of a group G by a normal subgroup H is a key source of abstract groups.
Subgroups
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If (G, *) is a group structure and S is a subset of G, S is a subgroup of G if (S, *) is a group structure and it meets the following criteria.
- Binary Structure: for each a, b S, ab S.
- Identity Existence: Assume e' S is such that e'a = a = ae' for every a S.
- Inverse Existence: There is an a1 S such that aa1 = e = a1a for any a S.
Things to Remember
- The natural language of describing the symmetries of a physical system is group theory.
- Group theory has many dimensions and applications both inside and outside of science and mathematics, and groups may be found in a variety of seemingly insignificant phenomena.
- The group is called a finite group if it has a limited number of components, with the number of components referred to as the group order.
- Subgroups are a subset of a group that is bounded by any of the group operations.
Sample Questions
Ques. How many properties can be held by a group? (3 marks)
Ans. The properties held by a group is:
- Closure
- Associative
- Commutative
- Identity element
- Inverse element
Ques. How will you show that a g-generated group is equivalent to a g-generated inverse group? (3 marks)
Ans. The cyclic subgroup created by an element g of a group G consists of all integer powers of g — where the identity element of G is defined as the zeroth power and negative powers of g are defined as the corresponding positive powers of g-1, the inverse of g. However, the cyclic subgroup created by g-1 is made up of all of its positive integer powers, which are all the negative integer powers of g; the identity element; and all negative integer powers of g-1, which are all the positive integer powers of g — all of which are the same set.
Ques. What are some applications of group theory? (3 marks)
Ans. Some applications of group theory are mentioned below:
- In Atomic and Molecular Spectroscopy, it is critical for determining the selection rules for spectroscopic transitions.
- It is used in chemistry to research and evaluate the symmetries and crystal structures of molecules, as well as various physical and chemical characteristics and spectroscopic features.
- It may be used in mathematics to classify identical symmetrical mathematical objects, such as geometric shapes (a circle is extremely symmetric and invariant under any rotation) and mathematical functions and operations.
- In Cryptography and Public Key techniques, the concepts of group, subgroups, and cosets are employed to ensure smooth data transfer.
- It has also been used in music screening in recent studies.
Ques. If G is a group and a, b are any two elements of G then prove (ab)-1 = b-1. a-1 (3 marks)
Ans. If a-1 and b-1 are the inverse of a and b respectively then,
a.a-1 = a-1.a = e
e is the identity element of G.
b.b-1 = b-1.b = e
Consider (ab).(b-1a-1) = a (b.b-1).a-1 [by associative law]
= a.e. a-1 [by inverse law]
= a . a-1 [by identity law]
= e [by inverse law]
Similarly, (b-1a-1)(ab) = b-1(a-1a)b [ by associative law]
= b-1(e)b [ by inverse law]
= b-1b [by identity law]
= e [ by inverse law]
Ques. How can I verify that ab = ba in a group G if (ab)2 = a2b2? (3 marks)
Ans. (ab)2 = (ab)(ab) = abab
And since we know that
(ab)2 = a2b2 = aabb (Use definition of squaring)
we get that
abab = aabb
a-1abab = a-1aabb
ebab = eabb
bab = abb
babb-1 = abb-1
bae = abe
ba = ab (Use the associativity and properties of inverse)
Ques. Assume that (G,) is a group and that a,b ∈ G, such that a2 = e and ab4a = b7. How can you prove that b33 = e? (4 marks)
Ans. ⇒ ab4 = b7a, b4a = ab7, ab7a = b4
⇒ b33 = (b7)3(b4)3 = ab4a ab4a ab4a ab7a ab7a ab7a = ab33a = (ab4a)8aba = b56aba
⇒ e = b23aba
⇒ b23 = ab-1a
⇒ b33 = ab33a = (ab7a)5(ab-1a)2 = b20b46 = b66
⇒ b66 = b33
⇒ b33 = e
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