Is S3 maximal subgroup in S4. As, I know if a subgroup is maximal then it should be normal isn't it? then is S3 normal in S4?
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5No, maximal subgroups of $G$ do not have to be normal in $G$. Yes, $S_3$ is a maximal subgroup if $S_4$, but it is not a normal subgroup. – Derek Holt Mar 21 '17 at 12:33
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All of the copies of $S_3$ are conjugate. Why do you think maximal subgroups are normal (they're not)? – Michael Burr Mar 21 '17 at 12:34
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@student A possible go to example when checking claims like, "maximal subgroups are normal", is to think that there are groups with no nontrivial normal subgroups, but every non-cyclic group has nontrivial maximal subgroup (by finite induction) – Pax Mar 21 '17 at 13:03
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@PaxKivimae;Is there any type of group, where maximal subgroups are always normal? Or any property to be normal? – student Mar 21 '17 at 15:10
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There are various conditions of when a subgroup is normal (other than the definition). For example, if it is a unique subgroup of a certain order. Another condition is if $[G:H]$ is the smallest prime dividing $|G|$ (see for example ) – Michael Burr Mar 21 '17 at 15:49
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thank you, I am expecting some property for maximal to be Normal. Anyway, the attached answer was nice. – student Mar 21 '17 at 15:56
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If $[G:H]$ is the smallest prime dividing $|G|$ is a special case of being maximal... – Michael Burr Mar 21 '17 at 18:23
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@MichaelBurr: yes, got it. Thank you – student Mar 21 '17 at 18:37