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For every finite $K$-approximate group $A$ in a group $G$, there are subgroups $H⊴C<G$ such that$C/H$ is a nilpotent group of class $O_K(1)$, and $A$ is contained in the union of $O_K(1)$ left cosets of $C$.

For a generator of a group $g$ of order $M$ and $x=g^y$, the function $f(a,b)=g^a{x}^{-b}$ hides the subgroup $\{(yb,b):b\in {\mathbb{Z}}_M\}$ of ${\mathbb{Z}}_M^2$. Each fiber is an affine coset $\{(yb+k,b):b\in {\mathbb{Z}}_M\}$. Measuring the function register gives its normalized coset state. Applying the positive-exponent quantum Fourier transform to both input registers gives amplitudeThe finite geometric series vanishes off $c_2=-y{c}_1$ and equals $M$ on that line. Thus each allowed output pair has probability $1/M$, independently of the coset offset. If $\gcd (c_1,M)=1$, a modular inverse recovers $y=-c_2{c}_1^{-1}$.

For the order-eight dihedral group generated by $r$ of order four and $s$ of order two with $srs=r^{-1}$, let $H=K=⟨s⟩$. There are exactly three double cosets: $H$, $H{r}^{2}H$, and $HrH$, of sizes two, two, and four. Their number does not divide eight. This differs from the number of ordinary cosets, which divides group order by Lagrange's theorem.

The doubling constant of a nonempty finite set $A$ is $|A+A|/|A|$. A small doubling constant indicates that $A$ behaves approximately like a coset of a subgroup.

For a function $f:X\to Y$ and $y\in Y$, its fiber is the set $f^{-1}(y)=\{x\in X:f(x)=y\}$. Fibers partition the domain after empty fibers are omitted. A function is injective exactly when every fiber has at most one element. Quantum hidden subgroup problem algorithms use functions whose fibers are cosets.

The connected component of the identity in a Lie group is an open normal subgroup. Connected coordinate neighbourhoods prove openness, and conjugation preserves the component of the identity. Every open identity neighbourhood generates it: the generated subgroup is open, its other cosets are open, and connectedness forces it to be the whole component.

For a finite group $G$ and subgroup $H$, the order of $H$ divides the order of $G$, and the quotient is the number of left cosets:

Every proper subgroup $H$ of a finite p-group is properly contained in its normalizer. Let $H$ act on the left cosets $G/H$ by left multiplication. The fixed cosets are those represented by elements of $N_G(H)$, so there are $[N_G(H):H]$ of them. All other orbits of a group action have sizes divisible by $p$. Since $[G:H]$ is divisible by $p$ and at least one coset is fixed, the fixed-coset count is a positive multiple of $p$. This also proves $[N_G(H):H]\ge p$.