Sylow not implies CDIN
This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., Sylow subgroup) need not satisfy the second subgroup property (i.e., CDIN-subgroup)
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- Sylow and TI implies CDIN: If we assume the additional condition that the Sylow subgroup intersects all its distinct conjugates trivially, then any two elements of it that are conjugate in the whole group are, in fact, conjugate in its normalizer.
Example of the symmetric group of degree four
Let be the symmetric group of degree four, say, on the set . Let be a -Sylow subgroup of , say:
Note that is a -Sylow subgroup of , and is a dihedral group of order eight. Consider the elements of given by:
- and are conjugate in : The element , for instance, conjugates to .
- : Since has index three in , either or . But the element , for instance, does not normalize . This forces .
- and are not conjugate in : In fact, is in the center of , and is not in the center of .