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The Ore conjecture, posed in 1951, states that every element of every finite non-abelian simple group is a commutator. Despite considerable effort, it remains open for various infinite families of simple groups. In this paper we develop new strategies, combining character-theoretic methods with other ingredients, and use them to establish the conjecture.

The other $P (G, V)$ -theorem

U. Meierfrankenfeld, B. Stellmacher (2006)

Rendiconti del Seminario Matematico della Università di Padova

The $p$ -nilpotency of finite groups with some weakly pronormal subgroups

Jianjun Liu, Jian Chang, Guiyun Chen (2020)

Czechoslovak Mathematical Journal

For a finite group $G$ and a fixed Sylow $p$ -subgroup $P$ of $G$ , Ballester-Bolinches and Guo proved in 2000 that $G$ is $p$ -nilpotent if every element of $P \cap G^{'}$ with order $p$ lies in the center of $N_{G} (P)$ and when $p = 2$ , either every element of $P \cap G^{'}$ with order $4$ lies in the center of $N_{G} (P)$ or $P$ is quaternion-free and $N_{G} (P)$ is $2$ -nilpotent. Asaad introduced weakly pronormal subgroup of $G$ in 2014 and proved that $G$ is $p$ -nilpotent if every element of $P$ with order $p$ is weakly pronormal in $G$ and when $p = 2$ , every element of $P$ with order $4$ is also...

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The lattice of very-well-placed subgroups

The Mathieu group $M_{12}$ and its pseudogroup extension $M_{13}$ .

The maximum distinguishing number of a group.

The maximum order of finite groups of outer automorphisms of free groups.

The Minimal 5-Representation of Lyon's Sporadic Group.

The number of conjugacy classes in a finite nilpotent group

The number of finite groups whose element orders is given.

The number of k-sums of abelian groups of order k

The Number of Solutions of x3 = 1 in a 3-Group.

The Number of Stem Covers of an Elementary Abelian p-Group.

The Ore conjecture

The other $P (G, V)$ -theorem

The $p$ -nilpotency of finite groups with some weakly pronormal subgroups

The $p^{*} p$ -injectors of a finite group

The partition problem for equifacetal simplices.

The periodic groups saturated by finitely many finite simple groups.

The pi-separability of Certain Factorizable Groups.

The primitive distance-transitive representations of the Fischer groups.

The product of generalized subnormal subgroups in finite groups.

The residually weakly primitive geometries of $J_{3}$ .

Currently displaying 41 – 60 of 89