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December 7th, 2016, 01:18 PM   #1
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No simple groups of size

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Last edited by skipjack; January 14th, 2017 at 08:03 PM.

 December 7th, 2016, 08:22 PM #2 Senior Member   Joined: Sep 2016 From: USA Posts: 635 Thanks: 401 Math Focus: Dynamical systems, analytic function theory, numerics Can you find an element of order 2? If so, it generates a subgroup of order 2, call this $H$. What can you say about $G/H$?
 January 10th, 2017, 11:40 AM #3 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 "When" is strange wording- 128 is always $2^7$!
 January 14th, 2017, 06:11 PM #4 Member   Joined: Jan 2016 From: Athens, OH Posts: 93 Thanks: 48 SDK's answer is confusing at best. Given a group G or order 128, it is not necessarily the case that any subgroup of order two is normal. So you need to know that any finite p-group (group of order a power of a prime p) has a non-trivial center. So if G is non-abelian, its center is a proper normal subgroup. So now you can assume G is abelian. By Cauchy's theorem there is a subgroup H of order 2; this H is now a proper normal subgroup.

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