Finite groups with a certain number of cyclic subgroups

In this short note, we describe the finite groups $G$ having $|G|-1$ cyclic subgroups. This leads to a nice characterization of the symmetric group $S_3$.Comment: accepted for publication in Amer. Math. Monthl

New lower bounds on subgroup growth and homology growth

We establish new strong lower bounds on the (subnormal) subgroup growth of a large class of groups. This includes the fundamental groups of all finite-volume hyperbolic 3-manifolds and all (free non-abelian)-by-cyclic groups. The lower bound is nearly exponential, which should be compared with the fastest possible subgroup growth of any finitely generated group. This is achieved by free non-abelian groups and is slightly faster than exponential. As a consequence, we obtain good estimates on the number of covering spaces of a hyperbolic 3-manifold with given covering degree. We also obtain slightly weaker information on the number of covering spaces of closed 4-manifolds with non-positive Euler characteristic. The results on subgroup growth follow from a new theorem which places lower bounds on the rank of the first homology (with mod p coefficients) of certain subgroups of a group. This is proved using a topological argument.Comment: 39 pages, 2 figures; v3 has minor changes from v2, incorporating referee's comments; v2 has minor changes from v1; to appear in the Proceedings of the London Mathematical Societ

Impartial avoidance games for generating finite groups

We study an impartial avoidance game introduced by Anderson and Harary. The game is played by two players who alternately select previously unselected elements of a finite group. The first player who cannot select an element without making the set of jointly-selected elements into a generating set for the group loses the game. We develop criteria on the maximal subgroups that determine the nim-numbers of these games and use our criteria to study our game for several families of groups, including nilpotent, sporadic, and symmetric groups.Comment: 14 pages, 4 figures. Revised in response to comments from refere

Cohomological Finiteness Conditions in Bredon Cohomology

We show that any soluble group $G$ of type Bredon-\FP_{\infty} with respect to the family of all virtually cyclic subgroups such that centralizers of infinite order elements are of type \FP_{\infty} must be virtually cyclic. To prove this, we first reduce the problem to the case of polycyclic groups and then we show that a polycyclic-by-finite group with finitely many conjugacy classes of maximal virtually cyclic subgroups is virtually cyclic. Finally we discuss refinements of this result: we only impose the property Bredon-\FP_n for some $n \leq 3$ and restrict to abelian-by-nilpotent, abelian-by-polycyclic or (nilpotent of class 2)-by-abelian groups.Comment: Corrected a mistake in Lemma 2.4 of the previous version, which had an effect on the results in Section 5 (the condition that all centralisers of infinite order elements are of type $FP_\infty$ was added