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Exhibit two distinct Sylow 2-subgroups of $S_5$ and an element of $S_5$ that conjugates one into the other.

Sketch of my answer:

$p=2,\, \alpha=3,\, m=15, n_2=1,3,5,15 $ and so the number of Sylow 2-subgroups of $S_5$ are $15.$ Two distinct Sylow 2-subgroups of $S_5$ are $<(12)>$ and $<(23)>.$ Observe that $(123)\in S_5$ and $(123)(12)(132)=(23).$ So $(123)$ is the element that conjugates one into the other.

Please comment/correct my answer.

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    anon- Thanks, that was silly of me!2012-10-05

1 Answers 1

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$K_1=<(1234), (24)>$

$K_2=<(1234), (34)>$.

$(23)K_1(23)=K_2$.