Let G be a finite group and let be subgroups of G such that:
(a) but for all
(b) The have parewise trivial intersections.
(c) The orders of the are prime numbers.
Show that .
I have tried induction. So, , where . Now, if I could prove that H is a subgroup of G, it is trivial that H, H1, ..., Hn satisfy (a), (b), (c). Therefore, . Furthermore,
using , if I could prove that , (but this I think I could prove it using the fact that H is a subgroup), I would have .
But unfortunately, I cannot prove that H is a group. Any suggestion will be welcome. Thanks for reading.