Lagrange's Theorem (Group Theory) Reference Cheat Sheet

Lagrange's Theorem is a central result in finite group theory that connects the size of a group to the size of each subgroup. This cheat sheet helps students organize the ideas of subgroup order, cosets, index, and element order. It is especially useful when proving divisibility statements, ruling out possible subgroups, and analyzing cyclic subgroups in finite groups.

The main theorem says that if $H \leq G$ and $G$ is finite, then $|H|$ divides $|G|$ . The quotient $[G:H] = \frac{|G|}{|H|}$ counts the number of left cosets of $H$ in $G$ . A key consequence is that the order of any element $a \in G$ divides $|G|$ , because $|a| = |\langle a \rangle|$ .

Lagrange's Theorem gives necessary conditions for subgroup orders, but it does not guarantee that every divisor of $|G|$ is the order of a subgroup.

Key Facts

If $G$ is a finite group and $H \leq G$ , then Lagrange's Theorem states that $|G| = [G:H]|H|$ .
The index $[G:H]$ is the number of distinct left cosets $gH$ of $H$ in $G$ .
Every left coset $gH$ has the same size as $H$ , so $|gH| = |H|$ for every $g \in G$ .
The distinct left cosets of $H$ partition $G$ , meaning every element of $G$ lies in exactly one left coset.
If $a \in G$ , then the order of $a$ satisfies $|a| = |\langle a \rangle|$ and therefore $|a|$ divides $|G|$ .
If $|G| = p$ for a prime $p$ , then $G$ is cyclic and its only subgroups have orders $1$ and $p$ .
For every $a \in G$ with $G$ finite, $a^{|G|} = e$ because $|a|$ divides $|G|$ .
Lagrange's Theorem implies possible subgroup orders must divide $|G|$ , but not every divisor of $|G|$ must occur as a subgroup order.

Vocabulary

Group: A group is a set $G$ with an operation that is closed, associative, has an identity element $e$ , and gives every element an inverse.
Subgroup: A subgroup $H \leq G$ is a subset of $G$ that is itself a group under the same operation.
Coset: A left coset of $H$ in $G$ is a set $gH = \{gh : h \in H\}$ for some $g \in G$ .
Index: The index $[G:H]$ is the number of distinct cosets of $H$ in $G$ .
Order of a Group: The order of a finite group $G$ is the number of elements in $G$ , written $|G|$ .
Order of an Element: The order of an element $a \in G$ is the smallest positive integer $n$ such that $a^n = e$ .

Common Mistakes to Avoid

Assuming every divisor gives a subgroup is wrong because Lagrange's Theorem only says subgroup orders must divide $|G|$ , not that every divisor of $|G|$ occurs.
Confusing $|G|$ , $|H|$ , and $[G:H]$ is wrong because they measure different things: group size, subgroup size, and number of cosets.
Forgetting that cosets partition $G$ is wrong because the proof of $|G| = [G:H]|H|$ depends on distinct cosets being disjoint and covering all of $G$ .
Thinking all cosets are subgroups is wrong because a coset $gH$ is a subgroup only in special cases, such as when $g \in H$ .
Using Lagrange's Theorem for infinite groups without care is wrong because the standard divisibility statement $|H| \mid |G|$ applies to finite groups.

Practice Questions

1 Let $G$ be a finite group with $|G| = 60$ and let $H \leq G$ with $|H| = 12$ . Find $[G:H]$ .
2 If $|G| = 35$ , list all possible orders of an element $a \in G$ according to Lagrange's Theorem.
3 Suppose $G$ is a finite group with $|G| = 18$ . Can $G$ have an element of order $8$ ? Explain using Lagrange's Theorem.
4 Explain why Lagrange's Theorem can rule out some subgroup orders but cannot prove that a subgroup of every divisor order exists.

Understanding Lagrange's Theorem (Group Theory) Reference

The reason behind the theorem is a counting argument based on translation. Start with a subgroup H and choose any element g of the larger group. Multiplying every element of H by g produces a coset.

This process does not lose or repeat elements. If two products are equal, multiplying by the inverse of g shows that the original subgroup elements were equal. Thus multiplication by g gives a perfect pairing between H and its coset.

Every coset therefore contains exactly as many elements as H. The important next step is that two cosets either have no shared elements or are actually the same set. This is what allows the whole group to be broken into equal sized blocks.

Symmetry groups give a concrete picture. Consider all eight symmetries of a square. The four rotations form a subgroup.

One coset is the rotation subgroup itself. A reflection followed by each possible rotation gives the other coset. These two collections contain four symmetries each, with no overlap, and together they include all eight symmetries.

This example shows that a coset need not be a subgroup. The reflection coset does not contain the identity symmetry, so it cannot be a subgroup. Cosets are translated copies used for counting, not necessarily smaller groups with their own identity.

Element order turns this counting result into a useful test for calculations. Repeatedly applying an element eventually returns to the identity in a finite group. The first positive number of repetitions that returns to the identity is the element order.

Since the powers of one element form a cyclic subgroup, that number must fit evenly into the size of the full group. This helps in modular arithmetic, where the nonzero residue classes that are relatively prime to a modulus form a group under multiplication. It explains why large powers can often be reduced to smaller ones.

It is important to check that the numbers being used are invertible modulo the modulus. All residues modulo a number do not form a multiplicative group when some have no inverse.

Students should treat divisibility as a filter, not as a construction method. If a proposed subgroup size does not divide the group size, the proposal is impossible immediately. If it does divide, more work is needed.

For example, the alternating group on four letters has twelve elements, yet it has no subgroup with six elements. Group structure matters, not just the total count. Another point to watch is the difference between left cosets and right cosets.

In a commutative group they match. In a noncommutative group they can differ. When they match for every group element, the subgroup is called normal, and this extra condition is needed before cosets can themselves be used to form a quotient group.

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