Let $k$ be a divisor of a finite group $G$ and $L_k(G)=\{x \in G \mid x^k=1\}$. Frobenius proved that the number $|L_k(G)|$ is always divisible by $k$. The following inverse problem is considered: for a given integer $n$, find all groups $G$ such that $\max \{k^{−1} |L_k(G)| \mid k \in \mathrm{Div}(G) \}=n$, where $\mathrm{Div}(G)$ denotes the set of all divisors of $|G|$. A procedure beginning with (in a sense) minimal members and deducing the remaining ones is outlined and executed for $n=8$.
Classification of finite groups via their breadth
Russo F
Primo
2020-01-01
Abstract
Let $k$ be a divisor of a finite group $G$ and $L_k(G)=\{x \in G \mid x^k=1\}$. Frobenius proved that the number $|L_k(G)|$ is always divisible by $k$. The following inverse problem is considered: for a given integer $n$, find all groups $G$ such that $\max \{k^{−1} |L_k(G)| \mid k \in \mathrm{Div}(G) \}=n$, where $\mathrm{Div}(G)$ denotes the set of all divisors of $|G|$. A procedure beginning with (in a sense) minimal members and deducing the remaining ones is outlined and executed for $n=8$.File in questo prodotto:
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