Counting conjugacy classes of subgroups in a finitely generated group

We prove by using simple number-theoretic arguments formulae concerning the number of elements of a fixed order and the number of cyclic subgroups of a direct product of several finite cyclic groups. We point out that certain multiplicative properties of related counting functions for finite Abelian groups are immediate consequences of these formulae.

This paper initiates a study into the contribution to the trace provided by the conjugacy classes.

In the paper new criteria of existence and conjugacy of Hall subgroups of finite groups are given.

Let $G$ be a finite group and $N<G$ a normal subgroup with $G/N$ abelian. We show how the conjugacy classes of $G$ in a given coset $qN$ relate to the irreducible characters of $G$ that are not identically $0$ on $qN$. We describe several consequences. In particular, we deduce that when $G/N$ is cyclic generated by $q$, the number of irreducible characters of $N$ that extend to $G$ is the number of conjugacy classes of $G$ in $qN$.

In [J. Algebra 452 (2016), 372-389], we characterise when the sequence of free subgroup numbers of a finitely generated virtually free group $\Gamma$ is ultimately periodic modulo a given prime power. Here, we show that, in the remaining cases, in which the sequence of free subgroup numbers is not ultimately periodic modulo a given prime power, the number of free subgroups of index~$\lambda$ in $\Gamma$ is - essentially - congruent to a binomial coefficient times a rational f...

Every finite group whose order is divisible by a prime $p$ has at least $2 \sqrt{p-1}$ conjugacy classes.

A packing of subsets $\mathcal S_1,..., \mathcal S_n$ in a group $G$ is a sequence $(g_1,...,g_n)$ such that $g_1\mathcal S_1,...,g_n\mathcal S_n$ are disjoint subsets of $G$. We give a formula for the number of packings if the group $G$ is finite and if the subsets $\mathcal S_1,...,\mathcal S_n$ satisfy a genericity condition. This formula can be seen as a generalization of the falling factorials which encode the number of packings in the case where all the sets $\mathcal S...

In this short note we give a formula for the number of chains of subgroups of a finite elementary abelian $p$-group. This completes our previous work [5].

The problem of finding the largest finite group with a certain class number (number of conjugacy classes), $k(G)$, has been investigated by a number of researchers since the early 1900's and has been solved by computer for $k(G) \leq 9$. (For the restriction to simple groups for $k(G) \leq 12$.) One has also tried to find a general upper bound on $|G|$ in terms of $k(G)$. The best known upper bound in the general case is in the order of magnitude $|G| \leq k(G)^{2^{k(G)-1}}$....

In 2000, L. H\'{e}thelyi and B. K\"{u}lshammer proved that if $p$ is a prime number dividing the order of a finite solvable group $G$, then $G$ has at least $2\sqrt{p-1}$ conjugacy classes. In this paper we show that if $p$ is large, the result remains true for arbitrary finite groups.