Groups and Subgroups - Examples and Constructions

Cosets provide a way to partition a group using a subgroup, leading to one of the most important results in finite group theory: Lagrange's theorem.

DefinitionLeft and Right Cosets

Let $H$ be a subgroup of $G$ and $g \in G$ . The left coset of $H$ by $g$ is: $gH = \{gh : h \in H\}$

The right coset of $H$ by $g$ is: $Hg = \{hg : h \in H\}$

Two left cosets $g_1H$ and $g_2H$ are either identical or disjoint. We write $g_1 \sim g_2$ if $g_1H = g_2H$ , which occurs if and only if $g_1^{-1}g_2 \in H$ .

ExampleCosets in $\mathbb{Z}$

Consider the subgroup $3\mathbb{Z} = \{\ldots, -6, -3, 0, 3, 6, \ldots\}$ in $\mathbb{Z}$ . The left cosets are:

$0 + 3\mathbb{Z} = \{\ldots, -6, -3, 0, 3, 6, \ldots\}$
$1 + 3\mathbb{Z} = \{\ldots, -5, -2, 1, 4, 7, \ldots\}$
$2 + 3\mathbb{Z} = \{\ldots, -4, -1, 2, 5, 8, \ldots\}$

These three cosets partition $\mathbb{Z}$ , corresponding to residue classes modulo 3.

The cosets of $H$ in $G$ partition $G$ into disjoint subsets, each having the same cardinality as $H$ . This observation leads directly to Lagrange's theorem.

TheoremLagrange's Theorem

Let $G$ be a finite group and $H$ a subgroup of $G$ . Then: $|G| = |H| \cdot [G:H]$

where $[G:H]$ is the index of $H$ in $G$ , equal to the number of distinct left (or right) cosets of $H$ in $G$ .

In particular, $|H|$ divides $|G|$ .

Remark

Lagrange's theorem has powerful consequences:

The order of any element divides the order of the group
If $|G| = p$ is prime, then $G$ is cyclic (hence isomorphic to $\mathbb{Z}_p$ )
For $g \in G$ , we have $g^{|G|} = e$ (generalizing Fermat's little theorem)

The converse of Lagrange's theorem is false: if $d$ divides $|G|$ , there need not exist a subgroup of order $d$ . For example, the alternating group $A_4$ has order 12 but no subgroup of order 6. However, for cyclic groups, the converse does hold: every divisor of $|G|$ corresponds to exactly one subgroup.

The index $[G:H]$ measures how many copies of $H$ fit into $G$ . When working with nested subgroups $K \leq H \leq G$ , we have the tower law: $[G:K] = [G:H] \cdot [H:K]$ .