Lagrange's Theorem

3.1 Cosets

Let $H \leq G$. For $a \in G$The left coset of $H$ containing $a$ is

$aH = \{ah : h \in H\}$

The right coset is $Ha = \{ha : h \in H\}$.

Proposition 3.1. The cosets of $H$ in $G$ partition $G$.

Proof. Define $a \sim b$ if $a^{-1}b \in H$. This is an equivalence relation: reflexive ($a^{-1}a = e \in H$), symmetric ($a^{-1}b \in H \Rightarrow b^{-1}a = (a^{-1}b)^{-1} \in H$), Transitive ($a^{-1}b, b^{-1}c \in H \Rightarrow a^{-1}c = (a^{-1}b)(b^{-1}c) \in H$). The equivalence class of $a$ is exactly $aH$. $\blacksquare$

Proposition 3.2. $|aH| = |H|$ for all $a \in G$.

Proof. The map $\phi : H \to aH$ given by $\phi(h) = ah$ is a bijection. $\blacksquare$

3.2 Lagrange’s Theorem

Theorem 3.3 (Lagrange’s Theorem). If $H$ is a subgroup of a finite group $G$Then $|H|$ divides $|G|$.

Proof. The cosets of $H$ partition $G$ into disjoint sets, each of size $|H|$. If there are $k$ cosets, Then $|G| = k \cdot |H|$So $|H|$ divides $|G|$. $\blacksquare$

The number of cosets is called the index of $H$ in $G$Denoted $[G : H]$.

Corollary 3.4. The order of any element of $G$ divides $|G|$.

Proof. $|g| = |\langle g \rangle|$And $\langle g \rangle \leq G$So $|g|$ divides $|G|$ by Lagrange. $\blacksquare$

Corollary 3.5 (Fermat’s Little Theorem). If $p$ is prime and $\gcd(a, p) = 1$Then $a^{p-1} \equiv 1 \pmod{p}$.

Proof. $\mathbb{Z}/p\mathbb{Z}$ has $p$ elements. The multiplicative group $(\mathbb{Z}/p\mathbb{Z})^*$ has order $p - 1$. The order of $[a]$ divides $p - 1$So $[a]^{p-1} = [1]$. $\blacksquare$

Corollary 3.6 (Euler’s Theorem). If $\gcd(a, n) = 1$Then $a^{\phi(n)} \equiv 1 \pmod{n}$Where $\phi$ Is Euler’s totient function.

3.3 Worked Example

Problem. Show that every group of prime order $p$ is cyclic.

Solution. Let $G$ be a group of order $p$ and $g \in G$ with $g \neq e$. By Corollary 3.4, $|g|$ divides $p$. Since $g \neq e$, $|g| \neq 1$. Since $p$ is prime, $|g| = p$. Thus $\langle g \rangle = G$And $G$ is cyclic. $\blacksquare$

3.4 Worked Examples: Computing Cosets

Problem. Let $H = \langle (1\ 2\ 3) \rangle \leq S_3$. Find all left cosets of $H$ in $S_3$.

Problem. Let $H = \langle 4 \rangle \leq \mathbb{Z}/12\mathbb{Z}$. Find all cosets of $H$.

3.5 Further Corollaries of Lagrange’s Theorem

Corollary 3.7. If $[G : H] = 2$Then $H \trianglelefteq G$.

Proof. There are exactly two left cosets $H$ and $aH$And exactly two right cosets $H$ and $Ha$. Since the cosets partition $G$We have $aH = G \setminus H = Ha$. Thus $gH = Hg$ for all $g \in G$ So $H$ is normal. $\blacksquare$

Corollary 3.8 (Product Formula). If $H, K \leq G$ are finite subgroups, then

$|HK| = \frac{|H||K|}{|H \cap K|}$

Proof. The map $H \times K \to HK$ given by $(h, k) \mapsto hk$ is surjective. For any $x = hk \in HK$ The fiber is $\{(hc^{-1}, ck) : c \in H \cap K\}$Which has size $|H \cap K|$. Thus $|H||K| = |HK| \cdot |H \cap K|$. $\blacksquare$

:::caution Common Pitfall The product $HK$ need not be a subgroup . However, $HK$ is always a subgroup when $H$ or $K$ is normal. In that case, $|HK| = |H||K|/|H \cap K|$ also divides $|G|$ by Lagrange.

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