A reference for the following result : Faithful irreducible representation over a field of characteristic zero implies cyclic centre.

Question

I was reading this page about the following result :

If a group has a faithful irreducible representation over any field of characteristic zero, then the center of the group is cyclic.

The said page has proof, but I am looking for a published book/paper, if available, for referring to.

Any help will be greatly appreciated.

Thanks in advance.

Nicky Hekster · Accepted Answer · 2020-03-02T08:59:01.380

1

This basically cycles back to the fact that a finite subgroup of the multiplicative group of a field of characteristic zero is cyclic. See I.M. Isaacs, Character Theory of Finite Groups, Lemma (2.27). And also here.

Let me explain it in more detail. First, if $\mathfrak{X}$ is a representation affording the character $\chi$, then $ker(\mathfrak{X})=ker(\chi)$. Here $ker(\mathfrak{X})=\{x \in G: \mathfrak{X}(x)=I\}$ and $ker(\chi)=\{x \in G: \chi(x)=\chi(1)\}$, see also Lemma (2.19) of Isaacs' book.

Now if $\chi$ is faithful then $ker(\chi)=\{1\}$. Hence Lemma (2.27) (d) and (f) will give you that $Z(G)$ must be cyclic.

edited Mar 02 '20 at 08:59

answered Mar 01 '20 at 12:24

Nicky Hekster

49,281

Thanks. Can you please explain a little more how to use Lemma 2.27 in this case? Sorry. – usermath Mar 01 '20 at 23:35
Edited into the answer. Hope you understand it now. – Nicky Hekster Mar 02 '20 at 08:59

A reference for the following result : Faithful irreducible representation over a field of characteristic zero implies cyclic centre.

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