 **This number is divisible by the order of the subgroup**  <http://arxiv.org/abs/1205.2824>.


The proof is short but non-trivial. Is this fact new or is it known for a long time?



The number of group elements whose squares lie in a given subgroup
In 2017, we learned that this fact was [proven by Shiro Iwasaki in 
1982](https://doi.org/10.1016/0021-8693(82)90093-X). 


