Applying combinatorial results to products of conjugacy classes

Abstract

Abstract Let

K =

x G

{K=x^{G}}

be the conjugacy class of an element x of a group G , and suppose K is finite.

We study the increasing sequence of natural numbers {

|

K n

|

}

n ≥ 1

{\{\lvert K^{n}\rvert\}_{n\geq 1}}

and consider restrictions on this sequence and the algebraic consequences.

In particular, we prove that if |

K 2

|

<

3 2

⁢

| K |

{\lvert K^{2}\rvert<\frac{3}{2}\lvert K\rvert}

or if

|

K 4

|

<

2 ⁢

| K |

{\lvert K^{4}\rvert<2\lvert K\rvert}

, then

K n

{K^{n}}

is a coset of the normal subgroup

[ x , G ]

{[x,G]}

for all

n ≥ 2

{n\geq 2}

or 4, respectively.

We then use these results to contribute to conjectures about the solubility of 〈 K 〉

{\langle K\rangle}

when

K n

{K^{n}}

satisfies certain conditions.