Let $G=\langle (1,2,3,4,5,6,7), (2,4,3,7,5,6) \rangle$.
I need to find sylow basis of $G$.
since the order of $(1,2,3,4,5,6,7)$ is 7 so we can prove that $n_7=1$ then it means that $\langle (1,2,3,4,5,6,7)\rangle\lhd G$ right? this is in the basis but how to I find the rest of the groups in the sylow basis?
I tried to find more groups in the basis, I thoght about the group $\langle (2,4,3,7,5,6)\rangle$ but she is from order 6 and we need from prime order. But I don't know how to find subgroups from order 2 and 3.
please help if you know how