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Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.

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Are there always more conjugacy classes in the kernel of a morphism to $Z_2$ than not?

There is a trivial example in which the number of conjugacy classes in the kernel and not in the kernel are equal. This is just the identity map $i:\mathbb{Z}_{2}\rightarrow\mathbb{Z}_{2}$.
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