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Group theory can be understood at a moderate level by high-school level students, and in fact well enough by interested undergraduate students for them to produce original research.

These properties are closure, associativity, identity, and inverse property. Groups serve as the building blocks of abstract algebra, they are more general and requisite that they appear in almost every branch of mathematics and the sciences. The concept of symmetry plays an important in group theory.

What is a group in maths? In maths, a group is the combination of a set and binary operation. For example, the set of integers with an addition operation forms a group and a set of real numbers with a binary operation; addition is also a group.

Group Theory Axioms and Proof 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) × b-1 × a-1= I, where I is the identity element of G. Consider the L.H.S of the above equation, we have, L.H.S = (a × b) × b-1 × b-1.

Symmetry is very important in chemistry researches and group theory is the tool that is used to determine symmetry. Usually, it is not only the symmetry of molecule but also the symmetries of some local atoms, molecular orbitals, rotations and vibrations of bonds, etc.

group theory, in modern algebra, the study of groups, which are systems consisting of a set of elements and a binary operation that can be applied to two elements of the set, which together satisfy certain axioms.

Some examples are as follows: Z \mathbb{Z} Z is cyclic, since it is generated by 1 {1} 1. This is because 1 + 1 = 2 1 + 1 = 2 1+1=2, 2 + 1 = 3 2 + 1 = 3 2+1=3, and so on, generating all positive integers.

The Order of a group (G) is the number of elements present in that group, i.e it's cardinality. It is denoted by |G|. Order of element a ∈ G is the smallest positive integer n, such that an= e, where e denotes the identity element of the group, and an denotes the product of n copies of a.

Group theory can be understood at a moderate level by high-school level students, and in fact well enough by interested undergraduate students for them to produce original research.

These properties are closure, associativity, identity, and inverse property. Groups serve as the building blocks of abstract algebra, they are more general and requisite that they appear in almost every branch of mathematics and the sciences. The concept of symmetry plays an important in group theory.

What is a group in maths? In maths, a group is the combination of a set and binary operation. For example, the set of integers with an addition operation forms a group and a set of real numbers with a binary operation; addition is also a group.

Group Theory Axioms and Proof 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) × b-1 × a-1= I, where I is the identity element of G. Consider the L.H.S of the above equation, we have, L.H.S = (a × b) × b-1 × b-1.

Symmetry is very important in chemistry researches and group theory is the tool that is used to determine symmetry. Usually, it is not only the symmetry of molecule but also the symmetries of some local atoms, molecular orbitals, rotations and vibrations of bonds, etc.

group theory, in modern algebra, the study of groups, which are systems consisting of a set of elements and a binary operation that can be applied to two elements of the set, which together satisfy certain axioms.

Some examples are as follows: Z \mathbb{Z} Z is cyclic, since it is generated by 1 {1} 1. This is because 1 + 1 = 2 1 + 1 = 2 1+1=2, 2 + 1 = 3 2 + 1 = 3 2+1=3, and so on, generating all positive integers.

The Order of a group (G) is the number of elements present in that group, i.e it's cardinality. It is denoted by |G|. Order of element a ∈ G is the smallest positive integer n, such that an= e, where e denotes the identity element of the group, and an denotes the product of n copies of a.