Conjugacy classes and finite pp-groups

Let $G$ be a finite $p$-group, where $p$ is a prime number, and $a\in G$. Denote by \Cl(a)=\{gag^{-1}\mid g\in G\} the conjugacy class of $a$ in $G$. Assume that |\Cl(a)|=p^n. Then \Cl(a)\Cl(a^{-1})=\{xy\mid x\in \Cl(a), y\in \Cl(a^{-1})\} is the union of at least $n(p-1)+1$ distinct conjugacy classes of $G$

Symmetric groups and conjugacy classes

Let S_n be the symmetric group on n-letters. Fix n>5. Given any nontrivial $\alpha,\beta\in S_n$, we prove that the product $\alpha^{S_n}\beta^{S_n}$ of the conjugacy classes $\alpha^{S_n}$ and $\beta^{S_n}$ is never a conjugacy class. Furthermore, if n is not even and $n$ is not a multiple of three, then $\alpha^{S_n}\beta^{S_n}$ is the union of at least three distinct conjugacy classes. We also describe the elements $\alpha,\beta\in S_n$ in the case when $\alpha^{S_n}\beta^{S_n}$ is the union of exactly two distinct conjugacy classes.Comment: 7 page

Homogeneous products of conjugacy classes

Let $G$ be a finite 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 ${\bf C}_G(a)={\bf C}_G(b)$. Then $a^G b^G$ is a conjugacy class if and only if $[a,G]=[b,G]=[ab,G]$ and $[ab,G]$ is a normal subgroup of $G$

Products of characters and derived length

Let G be a finite solvable group and \chi\in \Irr(G) be a faithful character. We show that the derived length of G is bounded by a linear function of the number of distinct irreducible constituents of $\chi\bar{\chi}$. We also discuss other properties of the decomposition of $\chi\bar{\chi}$ into its irreducible constituents.Comment: 12 pages, to appear J. Algebr