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Ionization energy is the energy required to remove an electron from a neutral atom, resulting in a positively charged ion. The ionization energy generally increases across a period (left to right) on the periodic table and decreases down a group (top to bottom). Electron affinity is the energy change that occurs when an electron is added to a neutral atom, forming a negative ion. Electron affinity values are generally more negative across a period (left to right) and become less negative or less exothermic down a group (top to bottom). A large ionization energy means an element is stable as a neutral species and doesn't want to lose an electron, whereas a highly exothermic electron affinity signifies an element's strong attraction towards an electron and the element's tendency to become a negatively charged ion.

The reason elements with large ionization energies have highly exothermic electron affinities is mainly due to the effective nuclear charge and shielding effect. As we move across a period from left to right, the atomic size decreases while the nuclear charge increases, making it harder for electrons to escape and the ionization energy to increase. On the other hand, a greater effective nuclear charge causes an increase in the attractive force between electrons and the nucleus, leading to highly exothermic electron affinities.

Now let's find the group of elements that could contradict the statement. By examining the periodic table, we notice that the elements that violate this statement are the Noble Gases (Group 18). They have very large ionization energies because of their stable full electron configuration, but their electron affinities are very low or even positive due to their stable closed-shell configuration requiring a lot of energy to add an electron. Therefore, Noble Gases do not follow the trend of having highly exothermic electron affinities along with their large ionization energies.