What are the steps in showing a group of order 30 is solvable/non-solvable?
I don't know how to proceed. All I know is that the group either has a group of order $5$ or $3$. I don't need all the steps for this problem, just an outline what to do.
What are the steps in showing a group of order 30 is solvable/non-solvable?
I don't know how to proceed. All I know is that the group either has a group of order $5$ or $3$. I don't need all the steps for this problem, just an outline what to do.
Some ideas:
1) Show that such a group always has one unique group of order 3 or one unique group of order 5
2) Using the above show that such a group always has a subgroup of order 15
3) Now use the following: if a group $\,G\,$ has a normal sbgp. $\,N\,$ s.t. both $\,N\,$ and $\,G/N\,$ are solvable, then $\,G\,$ is solvable
There are only four groups of order 30. They are $S_{3}\times \mathbb{Z}_{5}, D_{5}\times \mathbb{Z}_{3}, C_{30}, D_{15}$
$C_{30}$ is abelian so must be solvable. $S_{3}$ has a unique normal subgroup of order 3, and its quotient must be abelian. $D_{5}$ and $D_{15}$'s quotient with their normal subgroups must be abelian if you think of them as semidirect product. In fact, all groups of order 30 come as semidirect product of $\mathbb{Z}_{3}\times \mathbb{Z}_{5}$ with $\mathbb{Z}_{2}$. So they must be solvable. See this lecture note.