M.Sc Thesis | |

M.Sc Student | Ludar Nir |
---|---|

Subject | Group Characters with Many Values |

Department | Department of Mathematics |

Supervisor | PROF. David Chillag (Deceased) |

Given
a finite group G, and the set of its irreducible characters

Irr(G) = {*X*_{1}, *X*_{2},…,* X*_{n}},
the properties and values of *X*_{i} are driven from the
structure of G. On the other hand, it is known that the presence of some properties
and values of the characters of G affects the structure of G (even though it
does not determine it uniquely).

One
basic property of a character is being a class function: a character of G
accepts constant values on conjugacy classes of G. In this work we study groups
which have a special irreducible character *X*, which accepts different
absolute values on different conjugacy classes. i.e. if Cl_{G}(g) is
different then Cl_{G}(h) then |*X*(g)| is not equal to |*X*(h)|.

An immediate
consequence from the presence of such character in Irr(G) is the fact that G is
real, meaning that every character of G accepts only real values on the
elements of G. Another consequence is that G has an element *x* such that
the order of its centralizer, C_{G}(*x*), is less then 6.

According to those two facts, most of this work is devoted to the study of real groups which has an element that has a small centralizer. We separate the cases of solvable and non-solvable groups and map the structure of all the groups that has such character.