Lagrange's theorem: $\mathrm{\mid }G\mathrm{\mid }=[G:H]\mathrm{\mid }H\mathrm{\mid }$ for every subgroup $H$ of a finite group $G$

In a finite group, the subgroup, every coset and the set of cosets are finite

The left cosets of a subgroup partition the group

Every left or right coset of $H$ is equinumerous with $H$

The coset set $G\mathrm{/}H$ and the index $[G:H]$ of a subgroup

The order $\mathrm{\mid }G\mathrm{\mid }$ of a finite group and the order $\mathrm{ord}(g)$ of an element, with $\mathrm{ord}(g)=\mathrm{\infty }$ when no positive power of $g$ is the identity

The cardinality $\mid A\mid$ of a finite set

The sum rule: a finite disjoint union is finite with $\mid A\cup B\mid =\mid A\mid +\mid B\mid$ and $\mid {\bigcup }_{i\in I}{A}_i\mid ={\sum }_{i\in I}\mid A_i\mid$, and a sum over a finite index set splits along a partition

The sum ${\sum }_{i\in S}{a}_i$ over a finite index set, and its product form

Divisibility in $\mathbb{Z}$: $d\mid a$ when $a=dq$ for some integer $q$

The naturals embed in the integers