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Non-normal subgroups.

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Non-normal subgroups.

Hi there folks.

In the course of my research, I’ve been coming up with subgroups A of groups G with the property that the normaliser of A is the whole of G and the normal core of A is trivial. (The normal core of a subgroup is the largest subgroup of A which is normal in G, alternatively the intersection of all it’s conjugates).

I call these ”anti-normal” since they have the smallest possible core and the largest possible normaliser, as distinct from a normal subgroup which has the largest possible core and the smallest possible normaliser.

So my question is, have subgroups like these been looked at anywhere else? Is there a standard name I should be using? Is there much known about them? And if so, then is there any particularly good source where I could read about this?

Any help much appreciated. Cheers in advance.

Stevie Hair

Well, I have no trouble coming up with examples, but I can't recall any special name. :-(

(standard examples: any of the reflections in a dihedral group, any of the 2-cycles in a symmetric group, etc. Any time you have a set of generators which are all conjugates, and one of them is in your subgroup, will always be an example.)

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