Subgroups and Cosets

Entry conditions

You should already know the definition of a group and be comfortable with the idea of inverses.

Definitions

A subgroup $H$ of a group $G$ is a subset $H \subseteq G$ that is itself a group under the same operation. Equivalently, $H$ is a subgroup if and only if $H$ is non-empty, closed under the group operation, and closed under inverses.

Given a subgroup $H \leq G$ and an element $g \in G$, the left coset of $H$ by $g$ is $gH = \{g \cdot h \mid h \in H\}$. Right cosets $Hg$ are defined analogously.

Lagrange’s theorem: if $G$ is a finite group and $H \leq G$, then $|H|$ divides $|G|$. The number of distinct cosets is $|G|/|H|$, called the index of $H$ in $G$.

A subgroup $N$ is normal if $gN = Ng$ for all $g \in G$. When $N$ is normal, the cosets themselves form a group — the quotient group $G/N$.

Vocabulary (plain language)

• Subgroup: a group living inside a larger group.
• Coset: a “shifted copy” of a subgroup.
• Normal subgroup: a subgroup whose left and right cosets coincide.
• Quotient group: the group formed by the cosets of a normal subgroup.

Intuition

Cosets partition the group into equal-sized slices. Lagrange’s theorem says this partition is exact — no leftovers. Normal subgroups are exactly the subgroups where you can “divide out” and still get a well-defined group operation on the quotient.

Worked example

In $\mathbb{Z}$ under addition, the even integers $2\mathbb{Z}$ form a subgroup. The cosets are $2\mathbb{Z}$ (even numbers) and $1 + 2\mathbb{Z}$ (odd numbers). Since $\mathbb{Z}$ is abelian, every subgroup is normal, so the quotient $\mathbb{Z}/2\mathbb{Z}$ is a group with two elements — the integers modulo 2.

Common mistakes

• Assuming every subgroup is normal. In non-abelian groups, many subgroups are not normal.
• Confusing left and right cosets. They may differ when the subgroup is not normal.