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Show that a group of order $70$ can not be simple.

I've tried to solve using Sylow theorem. I got $1, 5, 7, 35$ Sylow $2$-subgroups, $1$ sylow $5$-subgroup and $1$ sylow $7$-subgroup. Now the only choice is $35$ Sylow $2$-subgroups which would yield $36$ elements. Now we are left with $34$ elements but we have only one sylow $5$-subgroup and one sylow $7$-subgroup.

Why all the elements of sylow subgroups are not adding up to $70$?

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    you know that 70=2.5.7 so use this fact2012-04-06

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The elements in a Sylow subgroup have a prime power as order, if the group has an element of order 35 (for example) it wont be in a sylow subgroup.

The exercise is already solved since you know that there is only one 7-sylow, it has to be normal (here is a proof).