Lagrange's Theorem In Abstract Math (Group Theory)

Lagrange's Theorem [Name of the Writer] [Name of the Institution] [Name of the Instructor] [Date] ABSTRACT Lagrange's theorem specifies that if a function is continuous in a closed interval [a, b], derivable in the open interval (a, b) and f (a) = f (b), then there is at least one point c between a and b for which f '( c) = 0. Lagrange's theorem has many applications in mathematical analysis, computational mathematics, and other fields. Lagrange’s Theorem Introduction Within the group theory, the outcome termed as Lagrange's Theorem specifies that for a finite group G, the order is divided by the order of any subgroup. This theorem was first originated that in the year 1770-71 with the intent of resolving the issue of solving the general polynomial equation of degree 5 or higher. Lagrange's lemma is one of the most general forms of the theorem and it specified that G is a finite group with subgroup H, then |H| divides |G|. * G can be partitioned by cosets of H * Each coset of H has the same order, |H|. * Hence |H| divides |G|. This clearly implies that a group consisting of 50 elements cannot acquire a subgroup of 8 elements for the reason that 8 does not evenly divide 50. On the other hand, the subgroup could have elements of 2,5,10 or 25 in the view of the fact that all these numbers are divisors of 50. It is eminent to mention here that the converse of Lagrange's theorem is not generally possible (Liu, 2003). This implies that if a number e is a divisor of the group that the group should acquire a subgroup of order “e”. Lagrange's theorem was introduced in the period of 1770s by one of the greatest mathematicians Joseph Louis Lagrange (1763-1813). Correspondingly, he was one of the first mathematicians to analyze the group structure. It is eminent to highlight here Lagrange being a student of physics and mathematics acquired special skills and talent pertaining to number theory. This talent and skills enabled him to devise a range of theories emphasizing on number theory and explicitly on group theory. This particular paper has focused on Lagrange's Theorem which is an important aspect of the group theory. This paper discussed different dimensions of the Lagrange's Theorem along with the proof by the means of sketches. It has also outlined the recent development of Lagrange's Theorem including cosets. Discussion In the context of the finite group theory, Lagrange's theorem is one of the fundamental theorems. According to this theorem, the order of the original group is divided by the order of the subgroup of the group. This assertion simply implies that the number of elements within a y subgroup ought to be divided evenly into the number of elements within the group. Often the Lagrange's theorem is referred to as Lagrange's lemma as well as Lagrange's group theorem. This section of the paper reviews the Lagrange's Theorem as well as its proof along with some other outcomes applicable to this study. Theorem A (Lagrange's Theorem): In this theorem suppose that G is a group of order n while H is the subgroup of G with the order of m. In such scenario, m is the divisor of n. In o...