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Feb 17, 2016 at 21:39 comment added Kıvanç Ersoy Thank you very much Professor Holt. Indeed, also I have realized that $A_{8}$ has two classes of involutions, the centralizer of the central involution is isomorphic to $C_{2}^{4} \rtimes A_{4}$ while the other involution has centralizer isomorphic to $D_{8}\times A_{4}$. Indeed, it is easy to see that my conjecture is false by this example too.
Feb 17, 2016 at 21:35 vote accept Kıvanç Ersoy
Feb 14, 2016 at 15:00 history answered Derek Holt CC BY-SA 3.0