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Finite groups whose set of numbers of subgroups of possible order has exactly 2 elements

Changguo Shao, Qinhui Jiang (2014)

Czechoslovak Mathematical Journal

Counting subgroups of finite groups is one of the most important topics in finite group theory. We classify the finite non-nilpotent groups $G$ whose set of numbers of subgroups of possible orders $n (G)$ has exactly two elements. We show that if $G$ is a non-nilpotent group whose set of numbers of subgroups of possible orders has exactly 2 elements, then $G$ has a normal Sylow subgroup of prime order and $G$ is solvable. Moreover, as an application we give a detailed description of non-nilpotent groups with...

Finite groups with exactly thirteen conjugacy classes

Antonio Vera López, Josu Sangróniz (1988)

Extracta Mathematicae

Finite groups with few self-normalizing subgroups

Huaguo Shi, Zhangjia Han (2012)

Colloquium Mathematicae

We describe finite groups which contain just one conjugate class of self-normalizing subgroups.

Finitely generated groups having a finite set of conjugacy classes meeting all cyclic subgroups

A. Ivanov (1999)

Colloquium Mathematicae

We study infinite finitely generated groups having a finite set of conjugacy classes meeting all cyclic subgroups. The results concern growth and the ascending chain condition for such groups.

Fixed points for positive permutation braids

Michał Misiurewicz, Ana Rodrigues (2012)

Fundamenta Mathematicae

Making use of the Nielsen fixed point theory, we study a conjugacy invariant of braids, which we call the level index function. We present a simple algorithm for computing it for positive permutation cyclic braids.