Hall subgroup - Imported Knowlpedia Content

Let $G$ be a finite group. A subgroup $H\le G$ is called a Hall subgroup if the order of $H$ is relatively prime to its index in $G$ , i.e.

\gcd(|H|,[G:H])=1,

where $\gcd$ denotes the greatest common divisor.

Hall subgroups generalize Sylow subgroups: if $|G|=p^n m$ with $p\nmid m$ and $P$ is a Sylow $p$ -subgroup, then $[G:P]=m$ is coprime to $|P|=p^n$ , so $P$ is a Hall subgroup. In finite solvable groups, Hall subgroups for prescribed sets of primes exist and enjoy conjugacy properties (Hall's theorem).

Examples

In $S_3$ (order $6$ ), any subgroup of order $3$ is a Hall subgroup (index $2$ ), and any subgroup of order $2$ is also a Hall subgroup (index $3$ ).
In $A_4$ (order $12$ ), the Klein four subgroup $V_4$ (order $4$ ) is a Hall subgroup because $[A_4:V_4]=3$ and $\gcd(4,3)=1$ .
Not every finite group has a Hall subgroup of every possible "coprime index" order; for instance, $A_5$ has no subgroup of order $15$ (so it has no Hall $\{3,5\}$ -subgroup).