lcmcharR
lcmcharR is a numerical invariant used in finite group theory to capture a coarse measure of the non-abelian character of a group. For a finite group R, let Irr(R) denote the set of irreducible complex characters of R, and let χ(1) be the degree of a character χ in Irr(R). The invariant lcmcharR is defined as the least common multiple of these degrees:
lcmcharR = lcm{ χ(1) : χ ∈ Irr(R) }.
In some literature this quantity is described as the least common multiple of the character degrees of
- Invariance: lcmcharR is preserved under group isomorphism; it depends only on the isomorphism class of R.
- Abelian criterion: If R is abelian, all irreducible characters have degree 1, so lcmcharR = 1. Conversely,
- Examples: For a cyclic group of order n, Irr(R) consists entirely of degree-1 characters, so lcmcharR =
- Computation is typically performed from the character table of the group, by extracting the degrees and
- As an invariant, lcmcharR provides a single-number summary of the distribution of character degrees and is
- It does not determine the group uniquely; many non-isomorphic groups share the same lcmcharR.
- Character theory, irreducible characters, character degree
- Finite groups, representation theory
Note: The term lcmcharR reflects a descriptive label used in some literature; it is not universally