Commuting fraction in quotient group is at least as much as in whole group

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Statement

Suppose G is a finite group and H is a normal subgroup. Let K = G/H be the quotient group. Then, the following are true:

  1. The Commuting fraction (?) (i.e., the fraction of pairs of elements that commute) in K is at least as much as in G.
  2. The Number of conjugacy classes (?) in K is bounded from below by the quotient of the number of conjugacy classes in G by the size of H.

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