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102 LearnersLast updated on September 26, 2025
Lagrange Theorem
According to Lagrange’s theorem, the order of any subgroup must divide the order of the group.
What is Lagrange’s Theorem?
According to Lagrange’s theorem, the order of a subgroup H divides the order of the group G. This relationship can be expressed as:
|G|=[G :H] .|H|
What is Coset?
A coset is the set of all products obtained by multiplying a fixed element of the group with each element of a subgroup.
In group theory, if g is an element of G, H is a subgroup of G, and G is a finite group, then
The left coset of H in G with respect to the element of G is g ∈ H is defined as:
gH ={gh \ h ∈ H}.
And,
The right coset of H in G with respect to the element g ∈ G is the set
Hg={gh \ h ∈ H}.
Let us now discuss the lemmas that support the Lagrange theorem.
Lemma 1: In a group G, every coset of a group H has the same number of elements as H, and there’s a one to one matching between them.
Lemma 2: The left coset relation, g1 ~ g2, is an equivalence relation if and only if, g1*H=g2*H, provided that G is a group with subgroup H.
Lemma 3: Assume that S is a set and that ~ is an equivalence relation on S. If A and B are two equivalence classes, then A ∩ B=Ø .
Lagrange Theorem Proof
The Lagrange statement is easily proven, with the help of the three lemmas. That is listed above.
Lagrange statement proof:
Let G be a finite group of order m, and let H be any subgroup of order n. Let us look how G can be divided into cosets of H.
Let’s now examine how each coset of gH is made up of n distinct elements.
Let H = {h1, h2, …, hn}, then ah 1, ah 2,..., ahn are the n different members of aH.
Assume that G satisfies the cancellation law: if ghi=ghj, then hi=hj.
The number of distinct left cosets, let’s say p, will likewise be finite since G is a finite group. Therefore, np, or the total number of elements of G, is the sum of the elements of all cosets. Thus, m=np
p = m/n
This shows that m, the order of the finite group G, divides n, the order of H.
Additionally, we observe that the index of a subgroup, which is, p divides the order of the group.
Thus, it was demonstrated that |G| = |H|
Lagrange Theorem Corollary
Corollary 1: The orders of any element a ∈ G divides the order of the group G, and in particular, am=e, if G is a group of finite order m.
Proof: Since a is the least positive integer and p is its order,
ap=e
Then we can state,
Group G’s elements, a,a2,a3, ..., ap-1, ap=e, are all unique and constitute a subgroup.
Since the element a creates a subgroup with p element and the whole group G also has p elements. That means the subgroup is actually the entire group. So, a generates the whole group.
Thus, we can write it as, If n is a positive integer, then write it as m=np.
Thus, the
am=anp=(ap)n=e
Thus, it is satisfying.
Corollary 2: A finite group G has no proper subgroups if its order is prime order.
Proof: Let us assume that group G has prime order m. The prime numbers property states that there are now only two divisors of m: 1 and m. Hence, the subgroups of G itself. As a result, no proper subgroups exist.
Corollary 3: A prime order group is cyclic if it has only two divisors.
Proof: Assume that a ≠ e ∈ G and that G is the prime order group of m.
The order of a is either 1 or m since it is a divisor of m.
However, since a ≠ e, the order of a, o(a) ≠ 1.
Consequently, the cyclic subgroup of G that is produced by a is of order m, a is the order of o(a)=p.
It establishes that G is cyclic, meaning that it is the same as the cyclic subgroup that a forms.
What is Lagrange's Mean Value Theorem?
According to Lagrange’s mean value theorem, if a function f satisfies the following conditions on the interval [a, b]:
On [a, b], f is continuous.
On (a, b), f is differentiable.
Consequently, at least one point c in (a, b) exists such that:
f'(c)=f (b) − f(a)b − a
This indicates that the derivative, or instantaneous rate of change, of the function equals the average rate of change over the interval at least once.
Real Life Applications on Lagrange Theorem
Lagrange theorem is used in our daily life for securing data, detecting errors and more. Let us examine some examples of Lagrange’s theorem’s applications
Cryptographic key Validation
By ensuring that the order of the encryption keys is a divisor of the group’s order, in systems such as elliptic curve cryptography, Lagrange’s theorem helps in the validation of encryption keys.
Molecular symmetry detection
Lagrange’s theorem is used by chemists to determine permitted molecular symmetries because it ensures that subgroup structures follow the symmetry rules of the molecule.
Designing fair puzzle and games
When designing puzzles, like Rubik’s cube, Lagrange’s theorem is used to make sure that each possible move creates a subgroup that splits the entire set of positions.
Robot motion control systems
Lagrange’s theorem guarantees that rotational or movement patterns in robotics belong to a subgroup of all possible movements, enabling predictable and repeatable actions.
Error-free data transmission
By ensuring that code groups behave like mathematical subgroups, we can apply algebraic structures that help in error detection and correction. Lagrange's theorem in coding theory facilitates the development of error-correcting codes.
Common Mistakes and How to Avoid Them in Lagrange Theorem
Students might make some common mistakes in Lagrange’s theorem. Let us examine those mistakes and see how to solve them correctly.
Mistake 1
Confusing “order” with position or arrangement
Instead of referring to the particular number of elements in a group or subgroup, students might think that the term of order describes how the elements are arranged or where they are in the list.
Always remember, never forget, that in group theory, the order of a group or subgroup indicates the number of elements it contains. For example,
A group P = {s, t, u, v}, has an order of 4, not the order of the elements listed.
Mistake 2
Assuming that every divisor gives a valid subgroup
Students may mistakenly assume that if a number divides the order of the group, then there must be a subgroup of that order.
Keep in mind that a subgroup of that size is not required just because a number divides the group’s order. For example, A4 contains 12 elements, but though the fact that 6 divides 12, it lacks a subgroup of order 6.
Mistake 3
Applying the theorem to infinite groups
Some students might mistakenly try to apply Lagrange’s theorem to groups that have an infinite number of elements, even though the theorem only works for finite groups.
Always know that, only finite groups are covered by Lagrange’s theorem. When there are an infinite number of elements in the group, it cannot be used.
Lagrange’s theorem does not apply to its subgroups because the set of integers ℤ under addition is infinite.
Mistake 4
Believing the theorem only applies to normal subgroups
Sometimes, students believe that Lagrange’s theorem only applies when the subgroup is normal.
Know that all subgroups, whether they are normal or not, are covered by Lagrange’s theorem. For example, the subgroup of order 2 or 5 satisfies the theorem if G is a group of order 10, even if it is not normal.
Mistake 5
Assuming left and right cosets are always equal
Students may make wrong decisions by assuming that the left and right cosets are the same.
Always know that the left and right cosets may differ. For example,
The left and right cosets of a subgroup in the symmetric group S3 may differ, Lagrange’s theorem still uses the number of cosets to determine the subgroup order.
Solved Problems on Lagrange’s Theorem
Problem 1
A group with an order of 10. Which orders might its subgroups fall under?
The divisors of 10 are possible subgroup orders:
{1, 2, 5, 10}
Explanation
Lagrange’s theorem tells us that the size of any subgroup should divide the size of the whole group. So, if a group K has 10 elements, its subgroups can have only 1, 2, 5, or 10 elements, it is because of those numbers that are evenly divided 10.
Problem 2
Can a group of order 30 have a subgroup of order 7?
No, 30 is not divided by 7.
Explanation
7 is not the divisor of 30, according to Lagrange’s theorem, which states that a subgroup’s order must divide the group’s order.
Problem 3
Order 60 belongs to group G. Is a subgroup of order 5 possible?
Yes, 5 divides 60.
Explanation
According to Lagrange’s theorem, any subgroup of order 60 can have order 5, since 5 divides 60
Problem 4
What is the only subgroup of order 1 that exists in a group G?
It is {e}, the trivial subgroup.
Explanation
Lagrange’s theorem is satisfied, since every group has a trivial subgroup made up of the identity elements, and its order is always 1.
Problem 5
Order 12 belongs to group L. Enumerate order that could exist.
{1, 2, 3, 4, 6, 12} are possible subgroup orders.
Explanation
The only acceptable subgroup orders are divisors of 12, as per Lagrange’s theorem, which says that the order of a subgroup must divide the order of the group.
FAQs On Lagrange’s Theorem
1.What is the state of the Lagrange’s theorem?
2.What is the value theorem of Lagrange?
3.What is Lagrange's theorem used for?
4.The formula for Lagrange’s theorem?
5.What is the Lagrangian principle of state?
6.How does learning Algebra help students in United States make better decisions in daily life?
7.How do technology and digital tools in United States support learning Algebra and Lagrange Theorem?
8.How can cultural or local activities in United States support learning Algebra topics such as Lagrange Theorem?
9.Does learning Algebra support future career opportunities for students in United States?
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
