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 R, and it may be denoted by alternative names such as LCD(R).

Basic properties

- 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, if lcmcharR > 1, the group is non-abelian.

- Examples: For a cyclic group of order n, Irr(R) consists entirely of degree-1 characters, so lcmcharR = 1. For the symmetric group S3, the irreducible degrees are 1, 1, and 2, giving lcmcharR = 2. For the alternating group A5, the degrees are 1, 3, 3, 4, and 5, so lcmcharR = lcm(1,3,4,5) = 60.

Computation and use

- Computation is typically performed from the character table of the group, by extracting the degrees and calculating their least common multiple.

- As an invariant, lcmcharR provides a single-number summary of the distribution of character degrees and is used as a coarse descriptor in comparative studies of finite groups.

- It does not determine the group uniquely; many non-isomorphic groups share the same lcmcharR.

See also

- 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 standardized across all sources.