Prime characters and factorizations of quasi-primitive characters
The authors introduce the notion of prime characters as well as of an admissible set of prime characters. By means of these concepts new results on quasi-primitive irreducible characters are obtained. We define these first. A character \(\chi\) of G is a prime character if \(\chi\) is non-linear, irreducible, quasi-primitive, \(M^*_ G(\chi)\) is homogeneous, and \(\chi_{F^*_ G(\chi)}\) is irreducible. [Here \(M^*_ G(\chi)\) is the following: \(M^*_ G(\chi):= F^*_ G(\chi)/Z_ G(\chi)\); \(Z_ G(\chi)/\ker(\chi):= Z(G/\ker(\chi))\); \(F^*_ G(\chi)/\ker (\chi):=F^*(G/\ker (\chi))\); \(F^*(G)\) is the generalized Fitting group of G.] \([M^*_ G(\chi)\) is the direct product of G-chief factors if \(\chi\) is quasi-primitive. Thus it makes sense to speak of the homogeneous component of a G-chief factor in \(M^*_ G(\chi)\). An irreducible character \(\eta\) of G is called quasi-primitive if \(\eta_ N\) is a multiple of some irreducible character of N for all normal subgroups N of G.] A set S of prime characters of G is said to be an admissible set of prime characters if for all characters \(\lambda\) and \(\rho\) belonging to S with \(\lambda\neq \rho\), \(M^*_ G(\lambda)\) and \(M^*_ G(\lambda)\) contain no G-isomorphic G-chief factors. Several theorems, corollaries and lemmas are obtained in this paper. We mention here two of the main results and two important theorems. By definition two characters \(\chi_ 1\) and \(\chi_ 2\) are associates if \(\chi_ 1=\chi_ 2\lambda\) where \(\lambda\) is a linear character. Theorem A. (i) If \(\{\rho_ 1,...,\rho_ m\}\) is an admissible set of prime characters of G, the \(\prod^{n}_{i=1}\rho_ i\) is a quasi- primitive irreducible character of G. (ii) Assume \(\chi\) is a quasi- primitive irreducible nonlinear character of G. There is an extension \((\hat G,\pi)\) of G such that \(\ker(\pi)\subseteq Z(\hat G)\cap \hat G'\) and \(\chi =\prod^{n}_{i=1}\rho_ i\) where \(\{\rho_ 1,...,\rho_ n\}\) is an admissible set of prime characters of \(\hat G.\) If \(\chi\) may also be written as \(\prod^{m}_{i=1}\beta_ i\) where \(\{\beta_ 1,...,\beta_ m\}\) is an admissible set of prime characters of \(\hat G,\) then \(n=m\) and an ordering may be chosen so that \(\beta_ i\) and \(\rho_ i\) are associates for \(i=1,...,n.\) Corollary C. Assume G has odd order and \(\chi\) is a primitive nonlinear irreducible character of G. Then there is an extension \((\hat G,\pi)\) of G such that \(\ker(\pi)\subseteq Z(\hat G)\cap \hat G'\) and \(\chi= \prod^{n}_{i=1}\rho_ i\) where each \(\rho_ i\) is a prime character of \(\hat G.\) If \(\chi= \prod^{m}_{i=1}\beta_ i\) where each \(\beta_ i\) is a prime character of \(\hat G,\) then \(n=m\) and an ordering may be chosen so that \(\rho_ i\) and \(\beta_ i\) are associates for \(i=1,...,n.\) Theorem 1.13. If \(\chi\) is a quasi-primitive nonlinear irreducible character of G, then there is an extension \(\hat G\) of G such that \(\chi =\prod^{n}_{i=1}\rho_ i\) where each \(\rho_ i\) is an absolutely tensor indecomposable prime character of \(\hat G.\) \(\hat G\) may be taken to be a central extension of G. Theorem 2.7. Assume \(\{\rho_ 1,...,\rho_ n\}\) is an admissible set of prime characters of G, then \(\chi =\prod^{n}_{i=1}\rho_ i\) is a quasi-primitive irreducible character of G. For further details and applications the reader is urged to study the paper itself.
absolutely tensor indecomposable prime character
scientific article; zbMATH DE number 3922936