Pronormality is not centralizer-closed

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This article gives the statement, and possibly proof, of a subgroup property (i.e., pronormal subgroup) not satisfying a subgroup metaproperty (i.e., centralizer-closed subgroup property).
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It is possible to have a group G and a pronormal subgroup H of G such that the centralizer C_G(H) is not pronormal.

Related facts


Example of the symmetric group

Further information: symmetric group:S4, dihedral group:D8

Suppose G is the symmetric group on the set \{ 1,2,3,4 \} and H is a 2-Sylow subgroup of G, given by:

H = \{ (), (1,2,3,4), (1,3)(2,4), (1,4,3,2), (1,3), (2,4), (1,2)(3,4), (1,4)(2,3) \}.

Then, H is a pronormal subgroup of G, because it is a Sylow subgroup and Sylow implies pronormal. On the other hand, we have:

C_G(H) = \{ (), (1,3)(2,4) \}.

This is not pronormal, because it is conjugate to the subgroup \{ (), (1,2)(3,4) \} via the permutation (2,3), but these are not conjugate in the subgroup they generate.

Let G be the symmetric group on the set \{ 1,2,3,4 \}.