Subgroup

A subgroup is a subset of a group that is itself a group under the restricted operation.

A Hall subgroup has order coprime to its index. For a set of primes $\pi$, a Hall $\pi$-subgroup has order using only primes of $\pi$ and index using only primes outside $\pi$. Thus its order contains the full prime-power contributions belonging to $\pi$. A Sylow subgroup is the single-prime instance. Hall subgroup existence in soluble groups generalizes Sylow existence to every prime set.

Every finite soluble group has a Hall subgroup for each prime set. Induct on the group order using an elementary abelian minimal normal subgroup. In the coprime hard case, lift a minimal normal subgroup of the quotient, choose its Sylow subgroup, and apply the Frattini argument. A proper normalizer reduces the order; a normal Sylow subgroup allows induction in its quotient. This proves existence without assuming an independent complement theorem.

The union of two subgroups is a subgroup exactly when one contains the other. Otherwise choose $h\in H\setminus K$ and $k\in K\setminus H$. Their product cannot lie in $H$, since $h^{-1}(hk)=k$ would then lie in $H$, and cannot lie in $K$, since $(hk)k^{-1}=h$ would then lie in $K$. The union therefore fails closure. Unlike the union, the subgroup generated by both subgroups always exists.

The index $[G:H]$ is the number of left cosets of $H$ in $G$.

Every subgroup $H\le G$ of index two is normal: the two left cosets and the two right cosets are both $H$ and its complement, so they agree.

A subgroup $H\le G$ has finite index when the coset space $G/H$ is finite. Intersecting a finite-index subgroup with a subgroup again gives a finite-index subgroup of the latter.