Related papers: z-Classes in finite groups of conjugate type (n,1)

We prove that any permutation group of degree $n \geq 4$ has at most $5^{(n-1)/3}$ conjugacy classes.

This paper concerns finite groups of class (at most) two and of odd prime exponent $p$. Such a group is called special if the center lies within its derived group. Every group of class 2 and exponent $p$ can be uniquely expressed as the…

Let $T$ be a finite non-abelian simple group. Giudici, Morgan and Praeger have shown that the order of $T$ is bounded above by a function depending on the maximum number of $\mathrm{Aut}(T)$-classes of elements of $T$ of prime-power order.…

In a group $G$, elements $a$ and $b$ are conjugate if there exists $g\in G$ such that $g^{-1} ag=b$. This conjugacy relation, which plays an important role in group theory, can be extended in a natural way to inverse semigroups: for…

We describe $\{2,3\}$-groups in which the order of a product of any two elements of orders at most $4$ does not exceed $9$ and the centralizer of every involution is a locally cyclic $2$-subgroup. In particular, we will prove that these…

We establish an explicit upper bound B(p,l,m), depending on p,l,m, on the number of conjugacy classes of order p^2 torsion elements u of type <l,m> of the Nottingham group defined over the prime field of characteristic p >0. In the cases…

In this paper, among other results, we give some sufficient conditions for every non-abelian subgroup of a group to be isoclinic with the group itself. It is also seen that under certain conditions, two groups have same number of element…

We characterize the group property of being with infinite conjugacy classes (or icc, i.e. \not= 1 and of which all conjugacy classes except 1 are infinite) for finite extensions of group.

The structure of finite and locally finite groups in which every element has prime power order (CP-groups) is well known. In this paper we note that the combination of our earlier results with the available information on the structure of…

Let k(B_0) and l(B_0) respectively denote the number of ordinary and p-Brauer irreducible characters in the principal block B_0 of a finite group G. We prove that, if k(B_0)-l(B_0)=1, then l(B_0)\geq p-1 or else p=11 and l(B_0)=9. This…

Let $G$ be a group. Write $G^{*}=G\setminus \{1\}$. An element $x$ of $G^{*}$ will be called deficient if $ \langle x\rangle < C_G(x)$ and it will be called non-deficient if $\langle x\rangle = C_G(x).$ If $x\in G$ is deficient…

Let $G$ be a nilpotent group and $a\in G$. Let $a^G=\{g^{-1}ag\mid g\in G\}$ be the conjugacy class of $a$ in $G$. Assume that $a^G$ and $b^G$ are conjugacy classes of $G$ with the property that $|a^G|=|b^G|=p$, where $p$ is an odd prime…

By using the structure and some properties of extraspecial and generalized/almost extraspecial $p$-groups, we explicitly determine the number of elements of specific orders in such groups. As a consequence, one may find the number of cyclic…

We prove that for any prime number $p$, every finite non-abelian $p$-group $G$ of class 2 has a noninner automorphism of order $p$ leaving either the Frattini subgroup $\Phi(G)$ or $\Omega_1(Z(G))$ elementwise fixed.

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…

This note collects several results on the capability of $p$-groups of class two and prime exponent. Among the new results, we settle the 4-generator case for this class.

In 2003, H\'{e}thelyi and K\"{u}lshammer proposed that if $G$ is a finite group and $p$ is a prime dividing the group order, then $k(G)\geq 2\sqrt{p-1}$, and they proved this conjecture for solvable $G$ and showed that it is sharp for those…

We classify all groups of order $p^5$ with non-trivial unramified Brauer groups. We show that if $p>3$, then there are precisely $\gcd (p-1,4)+\gcd (p-1,3)+1$ such groups.

Let $S$ be a semigroup. The elements $a,b\in S$ are called primarily conjugate if $a=xy$ and $b=yx$ for certain $x,y\in S$. The relation of conjugacy is defined as the transitive closure of the relation of primary conjugacy. In the case…

A group of order $p^n$ ($p$ prime) has an indecomposable polynomial invariant of degree at least $p^{n-1}$ if and only if the group has a cyclic subgroup of index at most $p$ or it is isomorphic to one of two particular groups of small…