Question
Asked Sep 22, 2019
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Show that S4
(a) has no normal subgroup of order three.

(b) has a normal subgroup of order four.

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Expert Answer

Step 1

To prove that (1) No normal subgroup of order 3 exists in S4 and (2) there does exist a normal subgroup of order 4 in the same group.

Step 2

(1) Any subgroup of order 3 must be generated by a 3-cycle. So we may assume that our subgroup H is of the form shown here.

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G=S Let H= {e. (123), (132)}. To show H is not normal in G.

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Step 3

We show that a suitable conjugate of an element in H is not in H. This shows that...

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G S Let H e, (123), (132)}. To show H is not normal in G. Consider the conjugate (14) (123)(14) (14)(123)(14) (234)H So, H not normal in S

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