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I have been trying to work this problem out for a bit and am stuck. Does anybody have any ideas how to proceed or solve this? I think this has something to do with rings and fields but I can't seem to find the connection.

EDIT: I know the other proof listed as a duplicate here, but it uses Cauchy's Theorem and we are not allowed to use that because we skipped that chapter.

user26857
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cambelot
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    See the answer here by Shubhodip Mondal: http://math.stackexchange.com/questions/225987/show-group-of-order-4n-2-has-a-subgroup-of-index-2 – Alex Wertheim Nov 12 '14 at 20:22

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Hint. Let $P$ be a Sylow $2$-subgroup, and consider $N_G(P)$. What is $\operatorname{Aut}(P)$ like? What does that mean about $N_G(P)/C_G(P)$? Use this to show that a $2$-complement is normal.

Alexander Gruber
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