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ON THE CODEGREES OF STRONGLY MONOLITHIC CHARACTERS OF FINITE GROUPS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 07 November 2024, pp. 1-7
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Let G be a finite group and let
$\chi $ be an irreducible character of G. The number 
$|G:\mathrm {ker}\chi |/\chi (1)$ is called the codegree of the character 
$\chi $. We provide several relations between the structure of G and the codegrees of the characters in a given subset of 
$\mathrm {Irr}(G)$, where 
$\mathrm {Irr}(G)$ is the set of all complex irreducible characters of G. For example, we show that if the codegrees of all strongly monolithic characters of G are odd, then G is solvable, analogous to the well-known fact that if all irreducible character degrees of a finite group are odd, then that group is solvable.
ON THE CHARACTERISATION OF SPORADIC SIMPLE GROUPS BY CODEGREES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 109 / Issue 1 / February 2024
- Published online by Cambridge University Press:
- 27 March 2023, pp. 57-66
- Print publication:
- February 2024
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Let G be a finite group and
$\mathrm {Irr}(G)$ the set of all irreducible complex characters of G. Define the codegree of 
$\chi \in \mathrm {Irr}(G)$ as 
$\mathrm {cod}(\chi ):={|G:\mathrm {ker}(\chi ) |}/{\chi (1)}$ and denote by 
$\mathrm {cod}(G):=\{\mathrm {cod}(\chi ) \mid \chi \in \mathrm {Irr}(G)\}$ the codegree set of G. Let H be one of the 
$26$ sporadic simple groups. We show that H is determined up to isomorphism by cod
$(H)$.
