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Certain Monomial Characters and Their Subnormal Constituents Carolina Vallejo Universitat de Val` encia St. Andrews, August 2013 Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 1 / 15

This is a joint work with G. Navarro. Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 2 / 15

Introduction Introduction Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 3 / 15

● Introduction Let ● be a group. A character ✤ ✷ ■rr✭ ● ✮ is said to be monomial if there exist a subgroup ❯ ✒ ● and a linear ✕ ✷ ■rr✭ ❯ ✮ , such that ✤ ❂ ✕ ● ✿ Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 4 / 15

Introduction Let ● be a group. A character ✤ ✷ ■rr✭ ● ✮ is said to be monomial if there exist a subgroup ❯ ✒ ● and a linear ✕ ✷ ■rr✭ ❯ ✮ , such that ✤ ❂ ✕ ● ✿ A group ● is said to be monomial if all its irreducible characters are monomial. Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 4 / 15

● ● Introduction There are few results guaranteeing that a given character of a group is monomial. Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 5 / 15

Introduction There are few results guaranteeing that a given character of a group is monomial. Theorem Let ● be a supersolvable group. Then all irreducible characters of ● are monomial. Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 5 / 15

Introduction There are few results guaranteeing that a given character of a group is monomial. Theorem Let ● be a supersolvable group. Then all irreducible characters of ● are monomial. Thus, supersolvable groups are monomial groups. Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 5 / 15

Introduction There are few results guaranteeing that a given character of a group is monomial. Theorem Let ● be a supersolvable group. Then all irreducible characters of ● are monomial. Thus, supersolvable groups are monomial groups. But this result depends more on the structure of the group than on characters themselves. Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 5 / 15

✤ ✷ ■rr✭ ● ✮ ● ✤ Introduction An interesting result. Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 6 / 15

Introduction An interesting result. Theorem (Gow) Let ● be a solvable group. Suppose that ✤ ✷ ■rr✭ ● ✮ takes real values and has odd degree. Then ✤ is rational-valued and monomial. Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 6 / 15

Introduction An interesting result. Theorem (Gow) Let ● be a solvable group. Suppose that ✤ ✷ ■rr✭ ● ✮ takes real values and has odd degree. Then ✤ is rational-valued and monomial. We give a monomiality criterium which also deals with fields of values and degrees of characters. Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 6 / 15

❥ ● ✭ P ✮ ✿ P ❥ ● ♣ P ✷ ❙②❧ ♣ ✭ ● ✮ ✤ ✷ ■rr✭ ● ✮ ♣ ♣ ✤ ◗ ❥ ● ❥ ♣ ✤ ♣ ❂ ✷ Introduction Notation: For ♥ an integer, we write ◗ ♥ ❂ ◗ ✭ ✘ ✮ ❀ where ✘ is a primitive ♥ th root of unity. Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 7 / 15

♣ ❂ ✷ Introduction Notation: For ♥ an integer, we write ◗ ♥ ❂ ◗ ✭ ✘ ✮ ❀ where ✘ is a primitive ♥ th root of unity. Theorem A Let ● be a ♣ -solvable group. Assume that ❥ N ● ✭ P ✮ ✿ P ❥ is odd, where P ✷ ❙②❧ ♣ ✭ ● ✮ for some prime ♣ . If ✤ ✷ ■rr✭ ● ✮ has degree not divisible by ♣ and the values of ✤ are contained in the cyclotomic extension ◗ ❥ ● ❥ ♣ , then ✤ is monomial. Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 7 / 15

Introduction Notation: For ♥ an integer, we write ◗ ♥ ❂ ◗ ✭ ✘ ✮ ❀ where ✘ is a primitive ♥ th root of unity. Theorem A Let ● be a ♣ -solvable group. Assume that ❥ N ● ✭ P ✮ ✿ P ❥ is odd, where P ✷ ❙②❧ ♣ ✭ ● ✮ for some prime ♣ . If ✤ ✷ ■rr✭ ● ✮ has degree not divisible by ♣ and the values of ✤ are contained in the cyclotomic extension ◗ ❥ ● ❥ ♣ , then ✤ is monomial. When ♣ ❂ ✷ , we can recover Gow’s result from Theorem A. Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 7 / 15

❆ ✻ Introduction The hypothesis about the index ❥ N ● ✭ P ✮ ✿ P ❥ is necessary. Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 8 / 15

❆ ✻ Introduction The hypothesis about the index ❥ N ● ✭ P ✮ ✿ P ❥ is necessary. For instance, the group SL(2,3) and the prime p=3. Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 8 / 15

❆ ✻ Introduction The hypothesis about the index ❥ N ● ✭ P ✮ ✿ P ❥ is necessary. For instance, the group SL(2,3) and the prime p=3. The solvability hypothesis is necessary in both Gow’s and Theorem A. Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 8 / 15

Introduction The hypothesis about the index ❥ N ● ✭ P ✮ ✿ P ❥ is necessary. For instance, the group SL(2,3) and the prime p=3. The solvability hypothesis is necessary in both Gow’s and Theorem A. The alternating group ❆ ✻ is a counterexample in both cases. Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 8 / 15

❇ ✙ Theory ❇ ✙ Theory Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 9 / 15

✭ ❛ ✮ ✤ ✭✶✮ ✙ ✭ ❜ ✮ ◆ ✴ ✴ ● ✤ ◆ ✙ ● ✙ ✙ ● ● ❇ ✙ Theory We say that ✤ ✷ ■rr✭ ● ✮ is a ✙ -special character of ● , if Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 10 / 15

✭ ❜ ✮ ◆ ✴ ✴ ● ✤ ◆ ✙ ● ✙ ✙ ● ● ❇ ✙ Theory We say that ✤ ✷ ■rr✭ ● ✮ is a ✙ -special character of ● , if ✭ ❛ ✮ ✤ ✭✶✮ is a ✙ -number. Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 10 / 15

● ✙ ✙ ● ● ❇ ✙ Theory We say that ✤ ✷ ■rr✭ ● ✮ is a ✙ -special character of ● , if ✭ ❛ ✮ ✤ ✭✶✮ is a ✙ -number. ✭ ❜ ✮ For every subnormal subgroup ◆ ✴ ✴ ● , the order of all the irreducible constituents of ✤ ◆ is a ✙ -number. Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 10 / 15

✙ ● ● ❇ ✙ Theory We say that ✤ ✷ ■rr✭ ● ✮ is a ✙ -special character of ● , if ✭ ❛ ✮ ✤ ✭✶✮ is a ✙ -number. ✭ ❜ ✮ For every subnormal subgroup ◆ ✴ ✴ ● , the order of all the irreducible constituents of ✤ ◆ is a ✙ -number. A B ✙ character of a group ● Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 10 / 15

❇ ✙ Theory We say that ✤ ✷ ■rr✭ ● ✮ is a ✙ -special character of ● , if ✭ ❛ ✮ ✤ ✭✶✮ is a ✙ -number. ✭ ❜ ✮ For every subnormal subgroup ◆ ✴ ✴ ● , the order of all the irreducible constituents of ✤ ◆ is a ✙ -number. A B ✙ character of a group ● may be thought as an irreducible character of ● induced from a ✙ -special character of some subgroup of ● . Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 10 / 15

❇ ✙ Theory We say that ✤ ✷ ■rr✭ ● ✮ is a ✙ -special character of ● , if ✭ ❛ ✮ ✤ ✭✶✮ is a ✙ -number. ✭ ❜ ✮ For every subnormal subgroup ◆ ✴ ✴ ● , the order of all the irreducible constituents of ✤ ◆ is a ✙ -number. A B ✙ character of a group ● may be thought as an irreducible character of ● induced from a ✙ -special character of some subgroup of ● . (True in groups of odd order). Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 10 / 15

Main results Main results Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 11 / 15

❇ ♣ ♣ Main results Theorem B Let ● be a ♣ -solvable group. Assume that ❥ N ● ✭ P ✮ ✿ P ❥ is odd, where P ✷ ❙②❧ ♣ ✭ ● ✮ for some prime ♣ . If ✤ ✷ ■rr✭ ● ✮ has degree not divisible by ♣ and its values are contained in the cyclotomic extension ◗ ❥ ● ❥ ♣ , then ✤ is a ❇ ♣ character of ● . Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters St. Andrews, August 2013 12 / 15