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A counterexample is an example that disproves a mathematical conjecture or a purported theorem. For example, the Peterson graph is a counterexample to many seemingly plausible conjectures in Graph Theory.
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Pronormaliser of a subgroup
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, …
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Characterisation of supersolvability of a finite group
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 …
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Subgroup embedding properties paranormality and polynormality
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 …