# Exposition of Lagrange’s Theorem

## Abstract

Group theory is one of the fundamental building blocks of abstract algebra and is also important for some sciences, such as chemistry and physics. The first book about group theory came out in 1870 which enabled a widespread understanding of theories that Galois created and spurred more research on the subject. Although the first book did not come out until 1870, there were many researchers before then who created theories that became the foundation for group theory up to a century beforehand. Lagrange was one of the first known researchers to provide information on group theory. He started by proving normal algebraic expressions up to the fourth degree. He then started to go into more theoretical mathematics (Mackey, 1973). Although his work was theoretical and was only proven by later mathematicians, it is still important in modern times for understanding group theory. Lagrange’s theories centered around solving solution sets for quantic equations using algebraic methods. In the late seventeen-hundreds, he created what is known as Lagrange’s Theorem (Kleiner, 1986). Lagrange’s Theorem states that the number of elements in a subgroup must divide the number of elements in the group. Through a poster presentation, I will define several concepts that have to do with functions, relations, groups and subgroups while providing proofs for propositions that follow those definitions. Through the explanation of these concepts, the building blocks needed to understand Lagrange’s Theorem will unfold. I will then prove Lagrange’s Theorem using the concepts that were described before. I will examine multiple examples of finite groups with the form of that illustrate Lagrange’s theorem.

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Exposition of Lagrange’s Theorem

Group theory is one of the fundamental building blocks of abstract algebra and is also important for some sciences, such as chemistry and physics. The first book about group theory came out in 1870 which enabled a widespread understanding of theories that Galois created and spurred more research on the subject. Although the first book did not come out until 1870, there were many researchers before then who created theories that became the foundation for group theory up to a century beforehand. Lagrange was one of the first known researchers to provide information on group theory. He started by proving normal algebraic expressions up to the fourth degree. He then started to go into more theoretical mathematics (Mackey, 1973). Although his work was theoretical and was only proven by later mathematicians, it is still important in modern times for understanding group theory. Lagrange’s theories centered around solving solution sets for quantic equations using algebraic methods. In the late seventeen-hundreds, he created what is known as Lagrange’s Theorem (Kleiner, 1986). Lagrange’s Theorem states that the number of elements in a subgroup must divide the number of elements in the group. Through a poster presentation, I will define several concepts that have to do with functions, relations, groups and subgroups while providing proofs for propositions that follow those definitions. Through the explanation of these concepts, the building blocks needed to understand Lagrange’s Theorem will unfold. I will then prove Lagrange’s Theorem using the concepts that were described before. I will examine multiple examples of finite groups with the form of that illustrate Lagrange’s theorem.