Subgroups and Cosets

Identifying and understanding the smaller, symmetric structures within a larger group is a fundamental technique in abstract algebra. Mastering subgroups and cosets allows you to decompose complex groups into manageable pieces, leading directly to powerful results like Lagrange’s Theorem, which imposes strict numerical constraints on what kinds of subgroups can exist. This knowledge is essential for analyzing symmetry in mathematics, physics, and cryptography.

Subgroups and How to Find Them

A subgroup is a subset $H$ of a group $G$ that is itself a group under the same operation as $G$. For a subset to be a subgroup, it must contain the identity element of $G$, be closed under the group operation, and contain the inverse of each of its elements. The trivial subgroup $\{e\}$ and the entire group $G$ itself are always subgroups.

Instead of checking all group axioms, we use the efficient subgroup test. A non-empty subset $H$ of a group $G$ is a subgroup if and only if for every $a, b \in H$, the element $a \cdot b^{-1}$ is also in $H$. This single condition verifies closure under the operation and under taking inverses, and guarantees the identity is present.

Consider the group of integers under addition, $G = (\mathbb{Z}, +)$. The set of even integers, $H = 2\mathbb{Z}$, is a subgroup. To apply the test, take any two even integers $a = 2m$ and $b = 2n$. Their "product" in additive notation is $a + (-b) = 2m - 2n = 2(m-n)$, which is also even and therefore in $H$. This confirms $H$ is a subgroup. In contrast, the set of positive integers is not a subgroup because it does not contain inverses (negatives) or the additive identity (0).

Cosets and Partitioning the Group

Once we have a subgroup $H$ of a group $G$, we can use it to slice the entire group into equal-sized pieces called cosets. For a fixed element $a \in G$, the left coset of $H$ containing $a$ is the set $aH = \{ a \cdot h \mid h \in H \}$. Similarly, the right coset is $Ha = \{ h \cdot a \mid h \in H \}$. In abelian (commutative) groups, left and right cosets are always identical, but in non-abelian groups they may differ.

The crucial property of cosets is that they partition the group $G$. This means every element of $G$ is in exactly one left coset (and exactly one right coset). Two left cosets $aH$ and $bH$ are either completely identical or entirely disjoint. They are identical precisely when $a^{-1}b \in H$ (or equivalently, when $b \in aH$).

Let's visualize this with a finite example. Let $G = S_3$, the symmetric group of permutations of three objects, with order 6. Let $H = \{ \text{id}, (1\;2) \}$, a subgroup of order 2. The left cosets of $H$ are:

• $\text{id} \cdot H = H = \{\text{id}, (1\;2)\}$
• $(1\;3) \cdot H = \{(1\;3), (1\;3\;2)\}$
• $(2\;3) \cdot H = \{(2\;3), (1\;2\;3)\}$

Notice each coset has exactly 2 elements, no element of $G$ appears in two different cosets, and the union of all cosets is $G$. This partition is the key to proving the next major theorem.

Lagrange's Theorem and Its Implications

Lagrange's Theorem is a cornerstone of finite group theory. It states: If $G$ is a finite group and $H$ is a subgroup of $G$, then the order (number of elements) of $H$ divides the order of $G$. The number of distinct left (or right) cosets of $H$ in $G$ is called the index of $H$ in $G$, denoted $[G : H]$. The theorem is captured by the equation: $$|G| = [G : H] \cdot |H|$$

Proof Outline: 1) The cosets of $H$ partition $G$. 2) There are exactly $[G:H]$ distinct left cosets. 3) Every left coset $aH$ has exactly $|H|$ elements (because the map $h \mapsto a \cdot h$ is a bijection from $H$ to $aH$). Since the cosets are disjoint and of equal size, the total number of elements in $G$ is the number of cosets times the size of each coset: $|G| = [G:H] \cdot |H|$. Therefore, $|H|$ must divide $|G|$.

This theorem has immediate, powerful consequences:

1. Order of an Element: The order of any element $a \in G$ (the smallest positive integer $n$ such that $a^n = e$) divides $|G|$. This is because the cyclic subgroup $\langle a \rangle = \{e, a, a^2, ...\}$ generated by $a$ has order equal to the order of $a$, and by Lagrange's Theorem, $|\langle a \rangle|$ divides $|G|$.
2. Groups of Prime Order: If $|G|$ is a prime number $p$, then $G$ must be cyclic. The only divisors of $p$ are 1 and $p$. Any non-identity element $a$ generates a subgroup of order greater than 1, which must therefore be of order $p$—meaning $G = \langle a \rangle$.
3. Fermat's Little Theorem (Group-Theoretic Glimpse): For a prime $p$, the multiplicative group of integers modulo $p$, $(\mathbb{Z}_p^*, \times)$, has order $p-1$. By consequence #1, for any integer $a$ not divisible by $p$, we have $a^{p-1} \equiv 1 \pmod{p}$, which is Fermat's result.

Common Pitfalls

1. Misapplying the Subgroup Test: A common error is checking only that $a \cdot b \in H$ for $a, b \in H$ (closure) but forgetting to check inverses separately. The standard subgroup test $a \cdot b^{-1} \in H$ efficiently combines both checks. For example, the set of natural numbers $\mathbb{N}$ is closed under addition within $(\mathbb{Z},+)$, but it is not a subgroup because it fails the inverse condition (and the $a \cdot b^{-1}$ test).
2. Confusing Elements and Cosets: An element and the coset it represents are different objects. In our $S_3$ example, the element $(1\;3)$ and the coset $(1\;3)H = \{(1\;3), (1\;3\;2)\}$ are not the same. Different representatives can define the same coset; $(1\;3\;2)$ and $(1\;3)$ are both in the same coset, so $(1\;3)H = (1\;3\;2)H$.
3. Assuming the Converse of Lagrange's Theorem: The theorem states that subgroup orders must divide the group order. The converse—"for every divisor $d$ of $|G|$, there exists a subgroup of order $d$"—is not generally true. The smallest counterexample is the alternating group $A_4$, which has order 12 but no subgroup of order 6.
4. Equating Left and Right Cosets: It is tempting to assume $aH = Ha$ for any element $a$. This is true if and only if the subgroup $H$ is normal in $G$. For non-normal subgroups, like our example $H = \{\text{id}, (1\;2)\}$ in $S_3$, the left and right cosets for a given element often differ.

Summary

• A subgroup $H$ of a group $G$ is a subset that forms a group under the same operation, efficiently verified using the subgroup test: for all $a, b \in H$, $a \cdot b^{-1} \in H$.
• Cosets ($aH$ or $Ha$) partition the group $G$ into equal-sized, disjoint subsets. Two cosets are equal if their representative elements are related by an element of the subgroup.
• Lagrange's Theorem mandates that for a finite group $G$ and subgroup $H$, the order of $H$ divides the order of $G$: $|G| = [G:H] \cdot |H|$.
• A critical consequence is that the order of any element in a finite group must divide the group's order.
• The converse of Lagrange's Theorem is false; the existence of a subgroup for every divisor of $|G|$ is not guaranteed.
• Understanding these structures provides the foundation for more advanced concepts like normal subgroups, quotient groups, and homomorphism theorems.