Linearly primitive implies cyclic-center

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Statement

Verbal statement

If a group has a faithful irreducible representation over complex numbers, then the center of the group is cyclic.

Property-theoretic statement

The group property of having a faithful irreducible representation over complex numbers (viz the property of being a linearly primitive group) implies the group property of having a cyclic center (viz the property of being a cyclic-center group).

Symbolic statement

Let G be a group with a faithful irreducible representation \rho over complex numbers. Then, Z(G) is a cyclic group.

Generalizations

It is in fact true that if a group has a faithful irreducible representation over any field of characteristic zero, then the center of the group is cyclic.

Proof

The proof follows from two facts.

Center goes to scalars

Schur's lemma states that for an irreducible representation \rho:G \to GL(V), the only elements of GL(V) that commute with everything in \rho(G) are the scalar matrices.

Since the image of any element in Z(G) must commute with everything in \rho(G), we conclude that the image of any element in Z(G) is scalar.

In particular, thus, we have a homomorphism from Z(G) to the multiplicative group of complex numbers (\mathbb{C}^*).

In particular, if \rho is faithful, Z(G) is embedded as a subgroup of \mathbb{C}^*.

Any finite subgroup of multiplicative group of complex numbers is cyclic

We know that the finite subgroups of the multiplicative group of complex numbers are precisely the cyclic groups of d^{th} roots of unity. These are always cyclic groups.

Combining these two facts, we obtain that the center is always cyclic.