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December 7th, 2016, 02:18 PM   #1
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No simple groups of size

Please help - what do I do when 128 is just 2 to the power 7?
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Last edited by skipjack; January 14th, 2017 at 09:03 PM.
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December 7th, 2016, 09:22 PM   #2
SDK
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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 $G/H$?
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January 10th, 2017, 12:40 PM   #3
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"When" is strange wording- 128 is always !
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January 14th, 2017, 07:11 PM   #4
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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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