Question

state and prove lagrang theorem on finite group
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Answer 1

Answer:

Lagrange Theorem

Lagrange theorem was given by Joseph-Louis Lagrange. Lagrange theorem states that in group theory, for any finite group say G, the order of subgroup H (of group G) is the divisor of the order of G i.e., O(G)/O(H). The order of the group represents the number of elements. In this lesson, let us discuss the statement and proof of the Lagrange theorem in Group theory. Lagrange theorem is one of the important theorems of abstract algebra. We will also have a look at the three lemmas used to prove this theorem with the solved examples. The understanding of a coset is also essential before you can fully understand the Lagrange theorem.

Lagrange Theorem Statement

Lagrange theorem states that the order of the subgroup H is the divisor of the order of the group G. This can be represented as;

|G| = |H|

Lagrange Theorem Proof

Let H be any subgroup with an order 'n' of a finite group G of order m. Let us consider the coset breakdown of G with respect to H. Now considering that each coset of aH comprises n different elements.

Let H = {h1,h2,…,hn}, then ah1, ah2,…,ahn are the n number of distinct members of aH.

Suppose,

ahi=ahj

⇒hi=hj be the cancellation law of G.

Now G is a finite group, so the number of discrete left cosets will also be finite, say p. So, the total number of elements of all cosets is np which is equal to the total number of elements of G.

Hence, m=np

p = m/n

This shows that n, the order of H, divides m i.e., is a divisor of m, the order of the finite group G. We also see that the index p is also a divisor of the order of the group.

Hence, proved, |G| = |H|