Powers in the wreath product of $G$ with $S_n$

Publication date: 10 October 2020

Abstract: In this paper we compute powers in the wreath product

G w r S_{n}

, for any finite group

G

. For

r g e q 2

, a prime, consider

o m e g a_{r} : G w r S_{n} o G w r S_{n}

defined by

g m a p s t o g^{r}

. Let

P_{r} (G w r S_{n}) = f r a c | o m e g a_{r} (G w r S_{n}) | | G |^{n} n!

, be the probability that a randomly chosen element in

G w r S_{n}

is a

r^{t h}

power. We prove,

P_{r} (G w r S_{n + 1}) = P_{r} (G w r S_{n})

for all

n o t e q u i v - 1 (e x t m o d r)

if, order of

G

is coprime to

r

. We also give a formula for the number of conjugacy classes that are

r^{t h}

powers in

G w r S_{n}

.

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