Counting conjugacy classes of subgroups in a finitely generated group

The authors classify the finite index subgroups of R. Thompson's group $F$. All such groups that are not isomorphic to $F$ are non-split extensions of finite cyclic groups by $F$. The classification describes precisely which finite index subgroups of $F$ are isomorphic to $F$, and also separates the isomorphism classes of the finite index subgroups of $F$ which are not isomorphic to $F$ from each other; characterizing the structure of the extensions using properties of the ...

Given a $p$-group $G$ and a subgroup-closed class $\mathfrak{X}$, we associate with each $\mathfrak{X}$-subgroup $H$ certain quantities which count $\mathfrak{X}$-subgroups containing $H$ subject to further properties. We show in Theorem I that each one of the said quantities is always $\equiv 1 \pmod p$ if and only if the same holds for the others. In Theorem II we supplement the above result by focusing on normal $\mathfrak{X}$-subgroups and in Theorem III we obtain a sharp...

We develop general formulae for the numbers of conjugacy classes and irreducible complex characters of finite p-groups of nilpotency class less than p. This allows us to unify and generalize a number of existing enumerative results, and to obtain new such results for generalizations of relatively free p-groups of exponent p. Our main tools are the Lazard correspondence and the Kirillov orbit method.

We establish several new bounds for the number of conjugacy classes of a finite group, all of which involve the maximal number c of conjugacy classes of a normal subgroup fixed by some element of a suitable subset of the group. To apply these formulas effectively, the parameter c, which in general is hard to control, is studied in some important situations. These results are then used to provide a new, shorter proof of the most difficult case of the well-known k(GV)-problem, ...

To every finitely generated group one can assign the conjugacy growth function that counts the number of conjugacy classes intersecting a ball of radius $n$. Results of Ivanov and Osin show that the conjugacy growth function may be constant even if the (ordinary) growth function is exponential. The aim of this paper is to provide conjectures, examples and statements that show that in "normal" cases, groups with exponential growth functions also have exponential conjugacy grow...

For suitable subgroups of a finitely generated group, we define the intersection number of one subgroup with another subgroup and show that this number is symmetric. We also give an interpretation of this number.

We extend and reformulate a result of Solomon on the divisibility of the title. We show, for example, that if $\Gamma$ is a finitely generated group, then $|G|$ divides $#\Hom(\Gamma,G)$ for every finite group $G$ if and only if $\Gamma$ has infinite abelianization. As a consequence we obtain some arithmetic properties of the number of subgroups of a given index in such a group $\Gamma$.

In "Subgroups of free profinite groups and large subfields of Q" (Israel J. Math. 39 (1981), no. 1-2, pages 25-45; MR 617288) A. Lubotzky and L. van den Dries raise the question whether a finitely generated, residually finite group is necessarily free if the rank function on its subgroups of finite index satisfies Schreier's well-known index rank relation (see Question 2 on p. 34). I answered this question in 1980 but, so far, I have not published my answer. This note fills t...

Let $G$ be a finite group and $H$ a subgroup of $G$. Each left transversal (with identity) of $H$ in $G$ has a left loop (left quasigroup with identity) structure induced by the binary operation of $G$. We say two left transversals are isomorphic if they are isomorphic with respect to the induced left loop structures. In this paper, we develop a method to calculate the number of isomorphism classes of transversals of $H$ in $G$. Also with the help of this we calculate the num...

In this note, we give a new formula for the number of cyclic subgroups of a finite abelian group. This is based on applying the Burnside's lemma to a certain group action. Also, it generalizes the well-known Menon's identity.