In my thesis I continue the study of Macdonald of $p$-groups of Frobenius Type, focusing on the set $\Irr(G|N)$ when $N=Z(G)$, as follows: First of all, with the support of professor Norberto Gavioli, I have found examples of p-groups $G$ such that $(G,Z(G))$ is a Camina pair, but $G$ is a not a Camina group. In my thesis there is not only the code, thanks to which it is possible to find such groups in the "SmallGroups" library of GAP; but also a detailed analysis of the found groups. Later, together with my tutor Carlo Maria Scoppola, I focused on the class mentioned above and then we have exploited the theory of characters of such groups, showing the following result: Let G be a $p$-group with $p>2$, $|Z(G)|=p^z$ and $z$ a non-negative integer. Then $(G,Z(G))$ is a Camina pair if and only if $|G| = p^{z + 2n}$ and there are $p^z -1 $ characters of degree $p^n$. As a consequence of previous theorem, we obtain: Let G be a $p$-group and $(G, K)$ be a C-pair. Then $|G : K|$ is an even power of $p$. As a consequence of previous result and assuming that the Conjecture of MacDonald (according to which if $(G, Z_2(G))$ is a C-Pair then $(G, Z(G))$ is a C-Pair too) is true, we obtain: Suppose that $|Z_2(G) : Z(G)| = p^n$, $|Z(G)|=p^z$ and $|G : Z_2(G)| = p^m$, if $(G,Z_2(G))$ is a Camina Pair, then $n$ is an even number. Note that in this case, the existence of $p^n - 1$ characters of degree $p^{\frac{m}{2}}$ would imply the existence of $p^z - 1$ characters of degree $p^{(\frac{m+n}{2})}$.

p-Groups of Frobenius Type / Velotto, Federica. - (2020 Jul 27).

p-Groups of Frobenius Type

VELOTTO, FEDERICA
2020-07-27

Abstract

In my thesis I continue the study of Macdonald of $p$-groups of Frobenius Type, focusing on the set $\Irr(G|N)$ when $N=Z(G)$, as follows: First of all, with the support of professor Norberto Gavioli, I have found examples of p-groups $G$ such that $(G,Z(G))$ is a Camina pair, but $G$ is a not a Camina group. In my thesis there is not only the code, thanks to which it is possible to find such groups in the "SmallGroups" library of GAP; but also a detailed analysis of the found groups. Later, together with my tutor Carlo Maria Scoppola, I focused on the class mentioned above and then we have exploited the theory of characters of such groups, showing the following result: Let G be a $p$-group with $p>2$, $|Z(G)|=p^z$ and $z$ a non-negative integer. Then $(G,Z(G))$ is a Camina pair if and only if $|G| = p^{z + 2n}$ and there are $p^z -1 $ characters of degree $p^n$. As a consequence of previous theorem, we obtain: Let G be a $p$-group and $(G, K)$ be a C-pair. Then $|G : K|$ is an even power of $p$. As a consequence of previous result and assuming that the Conjecture of MacDonald (according to which if $(G, Z_2(G))$ is a C-Pair then $(G, Z(G))$ is a C-Pair too) is true, we obtain: Suppose that $|Z_2(G) : Z(G)| = p^n$, $|Z(G)|=p^z$ and $|G : Z_2(G)| = p^m$, if $(G,Z_2(G))$ is a Camina Pair, then $n$ is an even number. Note that in this case, the existence of $p^n - 1$ characters of degree $p^{\frac{m}{2}}$ would imply the existence of $p^z - 1$ characters of degree $p^{(\frac{m+n}{2})}$.
p-Groups of Frobenius Type / Velotto, Federica. - (2020 Jul 27).
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11697/148691
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