General formula for number of elements of a given order in finite groups

Question

Is there a general formula to find the number of elements of a given order ?

For example, how many elements of order 5 are in a group with 15 elements? And if we try to find the number of elements of order 6 in S4 using Euler's totient function where S4 has 0 elements of order 6 can any one clarify when to use Euler's function and when not? Thank you

For $G=S_n$ see this duplicate. – Dietrich Burde Jan 05 '18 at 19:51 — Dietrich Burde, Jan 05 '18 at 19:51

score 2 · Accepted Answer · answered Jan 05 '18 at 19:28

2

There is no such a formula, because the number of elements of order $m$ of two groups $G$ and $H$ may be distinct even if $\#G=\#H$. For instance $\mathbb{Z}_4$ has $1$ element of order $4$, whereas $\mathbb{Z}_2\times\mathbb{Z}_2$ has none.

answered Jan 05 '18 at 19:28

José Carlos Santos

427,504

@Jose Carlos Santos what about Cn groups and Z/nZ can we always use Euler Function – Sara Jan 06 '18 at 10:21
@Sara Yes, of course. – José Carlos Santos Jan 06 '18 at 10:22

General formula for number of elements of a given order in finite groups

1 Answers1