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## Main Question or Discussion Point

The normal subgroup of a normal subgroup need not be normal in the original group (normalcy is not transitive). Could somebody provide me with an example of where this is the case? Thanks :D

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The normal subgroup of a normal subgroup need not be normal in the original group (normalcy is not transitive). Could somebody provide me with an example of where this is the case? Thanks :D

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mathwonk

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what examples have you tried? there should be some rather small suitable ones available.

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fresh_42

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Think of small symmetry groups or subgroups of them.

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fresh_42

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They are simple for ##n > 4## (and three). "Simple in" is a bit of a weird wording.Small symmetry groups.... Is the Alternating symmetry group (A_n) always simple in S_n?

If you consider a whole symmetric group it might be more difficult to prove because you have an additional transposition at hand to get closure under conjugation.If not that's the route i'm going to go.... It's always normal because the index in S_n is always going to be two obviously... Anyway if I find a normal subgroup of some A_n maybe it won't be normal in S_n? Do you think this is a smart route to take?

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So don't consider the whole symmetric group? What do you mean i'll have an additional transposition? an additional transposition compared to the alternating group? I haven't spent a whole lot of time tinkering around with the inner workings of the symmetric group, are there certain elements in most or all symmetric groups that are often times an obvious normal subgroup that I could look for non-transitive normal subgroups of?

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fresh_42

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Office_Shredder

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