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This article discusses the subgroup structure of the direct product ... A presentation for the group that we use is: G := \langle a,x,y \mid ... ...

A subgroup H of a group G is termed powering-invariant if it satisfies ... The dual property to this is quotient-powering-invariant subgroup ... ...

* Every group is transitively normal as a subgroup of itself. * The trivial subgroup is transitively normal in any group. ... ...

There is no standard name for this property, but both automorph-conjugate ... ! No. !! Shorthand !! A subgroup of a group is termed automorph-conjugate ... ...

A subgroup of a group is termed fully invariant or fully characteristic ... ! No. !! Shorthand !! A subgroup of a group is termed fully invariant ... ...

An element of a group is termed central if the following equivalent ... # Its centralizer is the whole group # It is the only element in its ... ...

of order eight and the cyclic maximal subgroup (the subgroup of order four ... The subgroup has index two and is hence a normal subgroup. Its left ... ...

Below are many equivalent definitions of characteristic subgroup. ... For a subgroup H \! of a group G \!, we denote the characteristicity ... ...

that we already have a group and a subgroup. To prove normality using ... For more definitions, see nonstandard definitions of normal subgroup. ... ...