CA group

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In mathematics, in the realm of group theory, a group is said to be a CA group or centralizer Abelian group if the centralizer of any nonidentity element is an Abelian subgroup.

CA groups were introduced in the context of the classification of finite simple groups. They occur in Suzuki's theorem which asserts that every finite simple nonAbelian CA group is of even order. This result is of course subsumed by the Feit-Thompson Theorem which states that every finite simple nonAbelian group is of even order.

Some facts about CA groups:

• Every CA group is a CN group.
• Any maximal Abelian subgroup of a CA group is a SA subgroup, viz, it has the property that it equals the centralizer of every nonidentity element in it.