nLab Artin-Lam induction exponent

Definition

Definition

For $G$ a finite group, its Artin-Lam induction exponent is the minimum $e(G)$ among positive natural numbers such that the $e(G)$-dimensional trivial representation is in the ideal of the rational representation ring of $G$ which is generated by $G$-representations that are induced by cyclic subgroups of $G$.

This definition according to the abstract of Madsen, Thomas & Wall 1983. Other authors take the smallest $e(G)$ such that $e(G) \cdot \chi$ is in this ideal for all rational representations $\chi$ (e.g. Yamauchi 70), in which case $e(G)$ is the exponent of the additive group underlying the quotient by the ideal, whence the terminology.

Related concepts

• free group actions on n-spheres

References

The original article:

• Tsit-Yuen Lam, Artin exponent of finite groups, Journal of Algebra 9 (1968) 94-119 (doi:10.1016/0021-8693(68)90007-0, pdf)

Further discussion:

• Kenichi Yamauchi, On the 2-part of Artin Exponent of Finite Groups, Science Reports of the Tokyo Kyoiku Daigaku, Section A Vol. 10, No. 263/274 (1970), 234-240 (jstor:43698746)

• K. K. Nwabueze, Some definition of the Artin exponent of finite groups (arXiv:math/9611212)

• S. Jafari and H. Sharifi, On the Artin exponent of some rational groups, Communications in Algebra, 46:4, 1519-152 (doi:10.1080/00927872.2017.1347665)

Textbook account:

• Charles Curtis, Irving Reiner, Methods of representation theory – With applications to finite groups and orders – Vol. I, Wiley 1981 (pdf)

Application to discussion of free group actions on n-spheres:

• Ib Madsen, Charles B. Thomas, C. T. C. Wall, Topological spherical space form problem III: Dimensional bounds and smoothing, Pacific J. Math. 106(1): 135-143 (1983) (pjm:1102721110)