group of order 80 is solvable How can we prove that group of order 80 is solvable without using Burnside's theorem (https://en.wikipedia.org/wiki/Burnside_theorem) ?

Since $80=2^4\cdot5$, any group of this order has either a unique Sylow $5$-subgroup or a unique Sylow $2$-subgroup (why? Count elements), so either way it has a normal Sylow subgroup, and either it has a quotient of order $2^4$ , which is a $2$-group and this solvable, or a quotient of order $5$ which is abelian and trivially solvable, so the whole group will be solvable.
