Insights on the structure of groups is often provided by the nature of the lattice of (certain types of) subgroups. Such issues go historically back to the original extension of the Mobius function from its number theoretic origins to a general lattice by the seminal work of Weisner and Hall. The paper listed below generalizes a classical theorem of Burnside, by working on the lattice of normal subgroups of a finite group.

An expression is obtained for the minimal number of conjugacy classes required to generate a group; this may be viewed as a generalization of the basis theorem of Burnside. The M\"obius function on the lattice of normal subgroups is then used to obtain equational relationships between the number of conjugacy classes in the major normal subgroups.

For conjugacy classes $C$ and $D$ we obtain an expression for $\sum\chi (C)\bar{\chi }(D),$ where the sum extends only over the faithful irreducible characters of a finite group. This expression is shown to be zero whenever the two conjugacy classes are distinct modulo the socle of the group. When $C$ and $D$ are the same class, and the class preserves its cardinality modulo the socle, we express the above sum in terms of certain invariants of the group.

The minimal number of conjugacy classes that generate a group