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Let $G$ be a finite group. The Wielandt subgroup of $G$ is defined to be the intersection of all the normalizers of the subnormal subgroups of $G$ i.e. $$w(G) = \bigcap_{H \triangleleft \triangleleft …

The pronormaliser of a subgroup $H$ in a group $G$, denoted $P_G(H)$, is defined to be the set of elements of $G$ that pronormalise $H$. That is, $$P_G(H) = \{g \in G \; | \; \exists x\in \langle H, …

Let $G = A \times B$. Suppose that $H \leq G$ such that $N_G(H) = N_A(\pi_A(H)) \times N_B(\pi_B(H))$ where $\pi_A$ and $\pi_B$ are the respective projection homomorphisms For simplicity and convenie …

Definition: A subgroup $H$ of $G$ is said to be pronormal in $G$ if for all $g\in G$, there exists $x \in \langle H, H^g \rangle$ such that $H^x =H^g$. Definition: A subgroup $H$ of $G$ is said to be …

Definition: Let $G$ be a finite solvable group and $\Sigma \in \text{H}(G)$, the set of Hall systems of $G$. The normaliser of $\Sigma$ is defined as $$ N_G(\Sigma) = \{ g\in G \,|\, H=H^g \text{ for …

A subgroup $H$ of $G$ is said to satisfy the Frattini Property if for any subgroup $K$ and $L$ such that $H\leq K \unlhd L$ implies that $L \leq N_L(H)K$. A subgroup is $H$ is pronormal in $G$ if for …

The pronormaliser of a subgroup $H$ in a group $G$, denoted $P_G(H)$, is defined to be the set of elements of $G$ that pronormalise $H$. That is, $$P_G(H) = \{g \in G \; | \; \exists x\in \langle H, …

Definition 1: A subgroup $H$ of a group $G$ is said to be abnormal in $G$ if for each $g\in G$, we have $g\in \langle H, H^g \rangle$. Definition 2: A finite group $G$ is called a $B$-group if every …

The following are subgroup embedding properties introduced by Bah and Borevich. Definition 1: A subgroup $H$ of $G$ is said to be paranormal if for each $g \in G$, we have that $H^{\langle H, H^g \ra …

Definition: A $T$-group is a group in which normality is a transitive relation. Definition: A subgroup $H \leq G$ is said to be weakly normal in $G$ if for each $g\in G$, $H^g \leq N_G(H)$ implies tha …

Let $G$ be a group. The pronormaliser of a subgroup $H$ in a group $G$, denoted $P_G(H)$, is defined to be the set of elements of $G$ that pronormalise $H$. That is, $$P_G(H) = \{g \in G \; | \; \exi …

In 1870, the American mathematician, Benjamin Pierce first introduced the term nilpotent in the context of his work on the classification of Algebras. In Algebra, an element $x$ of a ring $R$ is said …

Let $G = \mathrm{PSL}(3,q)$ for $q$ odd. I am trying to understand a question that involves understanding the subgroups that contain a Sylow $2$-subgroup, and in particular, are subgroups of odd index …

Note: Let $G$ be a finite group and $H \leq G$. Then it is clear that if $H \unlhd G$, then $\chi(h) \neq 0$ for irreducible constituents $\chi$ of the permutation character $(1_H)^G$ and $h\in H$. T …