Let $G$ be a group and $H$ be a normal subgroup of index $p$ ( a prime ) ; suppose $K$ is a subgroup of $G$ not contained in $H$ , then is it true that $G=HK$ ? ( I know that the fact is true if $p=2$ , but I don't know for the general case . Please help . Thanks in advance )
[Math] $G$ be a group and $H$ be a normal subgroup of index $p$ ( a prime ) ; suppose $K$ is a subgroup of $G$ not contained in $H$ , then $G=HK$
group-theorynormal-subgroups
Best Answer
By Lagrange's,
$$[G : HK][HK:H] = [G:H] = p$$
since $K \not\subset H$, $[HK:H] > 1 \implies [HK:H] = p$, so $[G:HK] = 1 \implies G = HK$.