Quotient group

Given a normal subgroup $𝑁 ⊴ 𝐺$ , the quotient group $𝐺 / 𝑁$ is a group of the cosets of $𝑁$ , defined as follows group

𝐺 / 𝑁 = {𝑔 𝑁 : 𝑔 \in 𝐺} (𝑔 1 𝑔 2) 𝑁 = (𝑔 1 𝑁) (𝑔 2 𝑁)

with the canonical projection

𝜋 : 𝐺 \to 𝐺 / 𝑁 𝑔 \mapsto 𝑔 𝑁

The quotient group is the natural application of the Algebraic quotient in the group context. However, instead of taking the quotient mod a congruence relation, it is typical to use the corresponding normal subgroup. Hence $𝑔 𝑁$ may alternatively be referred to as $[𝑔] 𝑁$ , taken to mean the equivalence class of $𝑔$ under the congruence induced by $𝑁$ . Another notation is to just use the elements of $𝐺$ but replace $=$ with $\equiv 𝑁$ .

Universal property

The quotient group with the canonical projection $(𝐺 / 𝑁, 𝜋)$ is characterized up to unique isomorphism by the universal property:

$𝑁 ⊴ k e r 𝜋$ . If $𝐻$ is a group and $𝜑 \in 𝖦 𝗋 𝗉 (𝐺, 𝐻)$ is a homomorphism with $𝑁 ⊴ k e r 𝜑$ , then there exists a unique homomorphism $―― 𝜑 \in 𝖦 𝗋 𝗉 (𝐺 / 𝑁, 𝐻)$ so that $𝜑 = ―― 𝜑 𝜋$ , i.e.

Proof

By construction, $𝜋 (𝑁) = {𝑒}$ . A homomorphism $𝜑 \in 𝖦 𝗋 𝗉 (𝐺, 𝐻)$ can be factored via $¯ 𝜑 𝜋 𝐴$ iff $𝜑 (𝑔 𝑁) = {𝜑 (𝑔)}$ , and this holds iff $𝜑 (𝑁) = {𝑒}$ . The uniqueness of $¯ 𝜑$ follows from $𝜋$ being an epimorphism: $𝜑 = ¯ 𝜑 𝜋 = 𝑓 𝜋 ⟹ 𝑓 = ¯ 𝜑$ . Therefore $(𝐺 / 𝑁, 𝜋)$ fulfils the universal property. If $(𝑄, 𝜓)$ also fulfils the universal property, then the following diagram commutes:

giving the required unique isomorphism.

Properties

By Lagrange’s theorem $| 𝐺 / 𝑁 | = (𝐺 : 𝑁) = | 𝐺 | | 𝑁 |$ .

Special quotients

Abelianization $𝐺 / [𝐺, 𝐺]$

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Table of Contents

Quotient group

Universal property

Properties

Special quotients

Graph View

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