How i can show that group of order 250 is a solvable group ?

|G| = 2*(5^3)

But i count show that (5^3)/{e} is abelian group

Thanks

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- Jan 1st 2010, 06:55 AMs.lateralussolvable group
How i can show that group of order 250 is a solvable group ?

|G| = 2*(5^3)

But i count show that (5^3)/{e} is abelian group

Thanks - Jan 1st 2010, 08:27 AMNonCommAlg
$\displaystyle P,$ the Sylow 5-subgroup of $\displaystyle G,$ is normal because $\displaystyle [G:P]=2$ and it's solvable because it's nilpotent. also $\displaystyle G/P$ is abelian and hence solvable. thus $\displaystyle G$ is solvable.

more generally and with the same argument we see that if $\displaystyle p < q$ are primes, then every group of order $\displaystyle pq^n, \ n \geq 0,$ is solvable. - Jan 1st 2010, 08:41 AMNonCommAlg
a more direct method: again let $\displaystyle P$ be the Sylow 5-subgroup of $\displaystyle G.$ we know that $\displaystyle P$ contains subgroups $\displaystyle P_1 \subset P_2$ such that $\displaystyle |P_1|=5, \ |P_2|=5^2.$ now look at this subnormal series:

$\displaystyle (1) \lhd P_1 \lhd P_2 \lhd P \lhd G.$ - Jan 1st 2010, 08:49 AMs.lateralus
How do you know that 5^2 is normal in 5^3 ?

- Jan 1st 2010, 09:21 AMDinkydoeCode:
`How do you know that 5^2 is normal in 5^3 ?`

There's a little theorem that states that any sub-group with index p is normal - Jan 1st 2010, 09:44 AMtonio