Cosets & Normal Subgroups

Cosets partition G; normal means gNg^-1 = N.

The idea

Given a subgroup $H$ of a group $G$, the cosets of $H$ partition $G$ into blocks of equal size, and when $H$ is normal the blocks themselves form a group, the quotient $G/H$.

The left coset of $g$ is the block $gH = \{gh : h \in H\},$ the set of elements differing from $g$ by an element of $H$. As proved in Subgroups & Lagrange, the cosets partition $G$ into $[G:H] = |G|/|H|$ blocks, each of size $|H|$.

To make the cosets into a group, we multiply them by the rule $(aH)(bH) = abH$. The danger is that a coset has many names: $aH = a'H$ whenever $a' = ah_1$ for some $h_1 \in H$. Computing the product from the names $a' = ah_1$ and $b' = bh_2$ gives $a'b' = a h_1 b h_2$, and this lies in the block $abH$ exactly when $h_1 b$ lands in $bH$ — that is, when $b^{-1} h_1 b \in H$. Requiring this for every element of $G$ and every element of $H$ is the condition $gHg^{-1} = H \text{ for every } g \in G,$ and a subgroup satisfying it is called normal. Normality makes the product of cosets independent of the names chosen, so it is precisely the condition under which the quotient group $G/H$ exists.

Ways to work on it

• Walkthrough. Cosets, the normality condition, and why it matters.
• Practice. Count cosets via the index.
• Hardest. Index-2 subgroups are always normal.

Not sure where to start? Take the ten-question placement test.