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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...

Free and non-free subgroups of the fundamental group of the Hawaiian Earrings

Andreas Zastrow (2003)

Open Mathematics

The space which is composed by embedding countably many circles in such a way into the plane that their radii are given by a null-sequence and that they all have a common tangent point is called “The Hawaiian Earrings”. The fundamental group of this space is known to be a subgroup of the inverse limit of the finitely generated free groups, and it is known to be not free. Within the recent move of trying to get hands on the algebraic invariants of non-tame (e.g. non-triangulable) spaces this space...

Geometric subgroups of surface braid groups

Luis Paris, Dale Rolfsen (1999)

Annales de l'institut Fourier

Let M be a surface, let N be a subsurface, and let n ≤ m be two positive integers. The inclusion of N in M gives rise to a homomorphism from the braid group B n N with n strings on N to the braid group B m M with m strings on M . We first determine necessary and sufficient conditions that this homomorphism is injective, and we characterize the commensurator, the normalizer and the centralizer of π 1 N in π 1 M . Then we calculate the commensurator, the normalizer and the centralizer of B n N in B m M for large surface braid...

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