Maximal among abelian subgroups

Symbol-free definition
A subgroup of a group is termed maximal among abelian subgroups or a maximal abelian subgroup or a self-centralizing abelian subgroup if it satisfies the following equivalent conditions:


 * 1) It equals its centralizer in the whole group.
 * 2) It is abelian and self-centralizing.
 * 3) It is abelian and is not properly contained in a bigger abelian subgroup.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed maximal among Abelian subgroups or a maximal abelian subgroup or a self-centralizing abelian subgroup if it satisfies the following equivalent conditions:


 * 1) $$H = C_G(H)$$, where $$C_G(H)$$ denotes the centralizer of $$H$$ in $$G$$.
 * 2) $$C(G)H \le H$$ and $$H$$ is abelian.
 * 3) $$H$$ is abelian, and if $$H \le K \le G$$ with $$K$$ Abelian, then $$H = K$$.

Stronger properties

 * In a nilpotent group or supersolvable group, maximal among abelian normal subgroups:

Weaker properties

 * Stronger than::c-closed self-centralizing subgroup
 * Stronger than::c-closed subgroup
 * Stronger than::Self-centralizing subgroup
 * Stronger than::Subgroup containing the center

Facts

 * In any group, there always exist maximal among abelian subgroups. In fact, every abelian subgroup of a group is contained in a maximal among abelian subgroup.
 * Two subgroups that are maximal among abelian subgroups need not be isomorphic. In fact, they may not even have the same size. For instance, in the symmetric group on three letters, there is a subgroup of order two and a subgroup of order three, both of them maximal among abelian subgroups.
 * In fact, any group can be expressed as a union of subgroups that are maximal among abelian subgroups. In particular, any non-abelian group has at least three distinct maximal among abelian subgroups.
 * In certain cases, any abelian subgroup can be replaced by a normal subgroup or 2-subnormal subgroup of the same size.