Group theory MOC

Centralizer in a group

The centralizer 𝐶𝐺(𝑎) of an element 𝑎 ∈𝐺 is a Subgroup of 𝐺 containing all elements that commute with 𝑎,¹ group i.e.

𝐶𝐺(𝑎)={𝑏∈𝐺∣𝑎𝑏=𝑏𝑎}

More generally, the centralizer 𝐶𝐺(𝑆) of any set 𝑆 ⊆𝐺 contains elements which commute with the whole of 𝑆, i.e.

𝐶𝐺(𝑆)={𝑥∈𝐺:(∀𝑠∈𝑆)[𝑥𝑠=𝑠𝑥]}

Proof of subgroups

Let 𝑎 ∈𝐺. Clearly 𝑒 ∈𝐶(𝑎), Given any 𝑏,𝑐 ∈𝐶(𝑎), clearly (𝑏𝑐)𝑎 =𝑏(𝑐𝑎) =𝑏(𝑎𝑐) =(𝑏𝑎)𝑐 =(𝑎𝑏)𝑐 =𝑎(𝑏𝑐), hence 𝐶(𝑎) is closed under the binary operation. Similarly, 𝑎𝑏 =𝑏𝑎 may be both pre- and postmultiplied by 𝑏−1 to obtain 𝑏−1𝑎 =𝑎𝑏−1, so 𝐶(𝑎) is closed under the inverse operation. Hence 𝐶(𝑎) is a subgroup of 𝐺 by Two step subgroup test.

Since the intersection of subgroups is a subgroup, 𝐶𝐺(𝑆) =⋂𝑠∈𝑆𝐶𝐺(𝑠) must also be a subgroup.

A related notion is the Centre of a group 𝑍(𝐺) =𝐶𝐺(𝐺), which includes only those elements that commute with all group elements.

Additional properties

• Clearly the centraliser itself need not be abelian, since the centraliser of any 𝑧 ∈𝑍(𝐺) will be the entire group. For example, in the non-abelian group 𝐷4, the centraliser 𝐶(𝑅180o̲) =𝐷4.

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Footnotes

1. 2017, Contemporary Abstract Algebra, p. 68 ↩