Cayley’s theorem states that every group \(G\) is isomorphic to a subgroup of the symmetric group acting on \(G\).
Multiplication by a member of \(G\) is a bijective function, as for each \(g\) there is also a \(g^{-1}\).
This means that multiplication of each member of \(G\) is a permutation, and so is a subset of the symmetric group on \(G\).
Lagrange’s theorem states that for any finite group \(G\), the order of every subgroup is a divisor of the order of \(G\).
Consider subset \(H\). We know that all cosets are disjoint, and that the union of all cosets is \(G\).
As cosets are the same size, we know that:
\(|G|=m|H|\), where \(m\) is the number of cosets.
This means that if a group has order \(10\), a subgroup must have order \(1\), \(2\) \(5\) or \(10\).