Lagrange's Theorem is a fundamental result in group theory. It states that for any finite group G and any subgroup H of G, the order of H must divide the order of G. Here we see a group G with 12 elements containing a subgroup H with 4 elements. Since 4 divides 12, this satisfies Lagrange's theorem.
The order of a group is simply the number of elements it contains. Similarly, the order of a subgroup is the number of elements in that subgroup. Lagrange's theorem tells us that these numbers have a special relationship. For example, if we have a group with 10 elements, we cannot have a subgroup with 3 elements, because 3 does not divide 10. This constraint is fundamental to group structure.
The proof of Lagrange's theorem relies on the concept of cosets. Cosets are formed by taking a subgroup H and multiplying each element by a fixed group element g. These cosets partition the entire group into disjoint subsets, each having exactly the same size as the original subgroup H. Here we see a group of 12 elements partitioned into 3 cosets, each containing 4 elements. The fundamental equation is: the order of G equals the index times the order of H.
Let's examine a concrete example. Consider the group Z₆, which consists of integers 0 through 5 under addition modulo 6. Take the subgroup H containing elements 0 and 3. We can form three cosets: H itself, 1 plus H which gives us 1 and 4, and 2 plus H which gives us 2 and 5. Each coset has exactly 2 elements, and we have 3 cosets total. Therefore, 6 equals 3 times 2, confirming that the order of H divides the order of Z₆.
Lagrange's theorem has profound applications throughout mathematics. It determines which subgroup sizes are possible for any given finite group, serving as a foundation for more advanced results like the Sylow theorems. The theorem is essential in group classification and has practical applications in cryptography and coding theory. As we can see, for a group of order 12, only subgroups of orders 1, 2, 3, 4, 6, and 12 are possible. This constraint reveals how group structure is fundamentally limited by divisibility relationships.