Subgroups and Lagrange's Theorem

1. What is a Subgroup?

Just as a vector space can contain smaller vector subspaces, a group can contain smaller groups. A subgroup is simply a subset of a group's elements that, using the same operation as the parent group, satisfies the four group axioms on its own.

Formally, a subset $H$ of a group $G$ is called a subgroup of $G$ if $H$ is itself a group under the operation of $G$. We denote this as $H\le G$.

While you could verify all four group axioms for any subset $H$, a more efficient method known as the subgroup test exists. For a non-empty subset $H$ of a group $G$, $H$ is a subgroup if and only if it satisfies two conditions:

1. Closure: For any two elements $h_1,h_2\in H$, their product $h_1{h}_2$ is also in $H$.
2. Inverses: For every element $h\in H$, its inverse $h^{-1}$ is also in $H$.

Associativity is inherited automatically from the parent group $G$, and the existence of the identity element is implied by these two conditions (since if $h\in H$, then $h^{-1}\in H$, and by closure, $h{h}^{-1}=e\in H$).

For a formal definition and some initial examples, the following reading is helpful.

Let's consider a few examples relevant to our previous lesson:

• Trivial Subgroups: For any group $G$, the set containing only the identity element, $\{e\}$, is always a subgroup. The group $G$ itself is also considered a subgroup of $G$.
• Rotations: Consider the group SO(3) of all rotations in three-dimensional space. The subset of all rotations about a single fixed axis (e.g., the z-axis) forms a subgroup. This subgroup is isomorphic to SO(2), which we studied in the last lesson.
• Permutations: Recall the symmetric group $S_3$, the group of permutations of three objects. Its order is $3!=6$. The subset corresponding to cyclic permutations, $\{e,(123),(132)\}$, forms a subgroup of order 3. Physically, this is the group of rotational symmetries of an equilateral triangle, often called $C_3$.

2. Cosets: Partitioning the Group

The key to unlocking Lagrange's theorem is the concept of a coset. A coset is formed by taking a subgroup $H$ and multiplying all of its elements by a single element $g$ from the larger group $G$. This procedure effectively creates "shifted copies" of the subgroup.

For a subgroup $H\le G$ and an element $g\in G$, we define:

• The left coset of $H$ with respect to $g$ as the set $gH=\{gh:h\in H\}$.
• The right coset of $H$ with respect to $g$ as the set $Hg=\{hg:h\in H\}$.

For now, we'll focus on left cosets, but all results apply equally to right cosets. A crucial point is that cosets are generally not subgroups themselves, as they typically do not contain the identity element.

The following video provides an excellent intuitive picture of how these cosets "tile" the entire group, which is the foundation for the proof of Lagrange's theorem.

The video brilliantly demonstrates the three essential properties of cosets:

1. Equal Size: Every coset $gH$ has the same number of elements as the subgroup $H$.
2. Disjointness or Identity: Two left cosets $g_{1}H$ and $g_{2}H$ are either identical or have no elements in common.
3. Completeness: The union of all distinct left cosets of $H$ is the entire group $G$.

In essence, a subgroup $H$ allows us to partition the group $G$ into a collection of disjoint sets (the cosets), each having the same size as $H$.

3. Lagrange's Theorem

With the understanding that a subgroup $H$ and its cosets partition the parent group $G$ into disjoint blocks of equal size, Lagrange's theorem follows almost as a direct observation.

Lagrange's Theorem: If $G$ is a finite group and $H$ is a subgroup of $G$, then the order of $H$ (the number of elements in $H$) divides the order of $G$.

We can write this as:

where $|G|$ and $|H|$ are the orders of the groups, and $l$ is a positive integer. This integer $l$ is called the index of $H$ in $G$, written as $[G:H]$, and it corresponds to the number of distinct cosets of $H$ in $G$.

The following resources formalize this theorem and its derivation.

Implications of Lagrange's Theorem

Lagrange's theorem is remarkably powerful despite its simple statement. It places a strong constraint on the possible substructures of a finite group.

• Order of Elements: The order of an element $g$ is the smallest positive integer $n$ such that $g^n=e$. The set $\{g,g^2,\dots ,g^n=e\}$ forms a cyclic subgroup of order $n$. Therefore, as a direct corollary of Lagrange's theorem, the order of any element of a finite group must be a divisor of the order of the group.
• Groups of Prime Order: If a group $G$ has a prime order $p$, its only possible subgroups are the trivial ones: $\{e\}$ (order 1) and $G$ itself (order $p$). This implies that any element $g\ne e$ must have order $p$, and the cyclic subgroup it generates, $⟨g⟩$, must be the entire group $G$. Therefore, any group of prime order is cyclic.

Let's illustrate these ideas with a concrete example. Consider the group $G={\mathbb{Z}}_8=\{0,1,2,3,4,5,6,7\}$ under addition modulo 8. Its order is $|G|=8$. Let's choose the subgroup $H=\{0,4\}$, which has order $|H|=2$.

According to Lagrange's theorem, the index of $H$ in $G$ must be $[G:H]=|G|/|H|=8/2=4$. This means we should be able to partition $G$ into 4 distinct cosets. Let's find them (using additive notation, so $g+H$):

• $0+H=\{0+0,0+4\}=\{0,4\}=H$
• $1+H=\{1+0,1+4\}=\{1,5\}$
• $2+H=\{2+0,2+4\}=\{2,6\}$
• $3+H=\{3+0,3+4\}=\{3,7\}$

If we try to generate another coset, say with 4, we find $4+H=\{4,8(\mathrm{mod}8)\}=\{4,0\}$, which is the same as $H$. The four cosets we found, $\{0,4\},\{1,5\},\{2,6\},\{3,7\}$, are disjoint and their union is all of $G$.

An Important Caveat

It is crucial to note that the converse of Lagrange's theorem is false. Just because an integer $k$ divides the order of a group $G$, it does not guarantee that a subgroup of order $k$ exists. The smallest counterexample is the group $A_4$ (the group of even permutations of 4 objects), which has order 12 but contains no subgroup of order 6.

Conclusion

In this lesson, we dissected the internal structure of groups by introducing subgroups and cosets. This led us to one of the most fundamental results in finite group theory, Lagrange's theorem.

Here are the key takeaways:

• A subgroup is a subset of a group that is also a group under the same operation.
• Cosets are "shifted copies" of a subgroup, $gH$, which allow us to partition a group $G$ into disjoint sets of equal size.
• Lagrange's Theorem states that the order of a subgroup must divide the order of the group, $|G|/|H|=[G:H]$.
• This theorem implies that the order of any element must also divide the order of the group, and that any group of prime order must be cyclic.

We have now seen how to partition a group using a subgroup. This raises a natural question: can the set of cosets itself form a group? The answer is "sometimes," but it requires the subgroup to have a special property called "normality". In our next lesson, we will explore a different way to partition a group's elements—into conjugacy classes—which will lead us directly to this concept of normal subgroups and the construction of new groups called quotient groups.