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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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Probably because your group might have elements of order 14 for example, which don't belong to any Sylow subgroup... –  N. S. Apr 6 '12 at 19:03
    
Probably there are elements of composite order. –  user21436 Apr 6 '12 at 19:04
    
BTW, you may want to edit your title. Your question seems to be about something connected with but not exactly about simpleness of groups. –  user21436 Apr 6 '12 at 19:06
    
you know that 70=2.5.7 so use this fact –  Babak Miraftab Apr 6 '12 at 19:07

1 Answer 1

up vote 5 down vote accepted

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).

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Thank you very much. –  Faisal Apr 6 '12 at 19:23

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