Commutator

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.

Group theory

The commutator of two elements, g and h, of a group G, is the element

It is equal to the group's identity if and only if g and h commute (i.e., if and only if gh = hg). The subgroup of G generated by all commutators is called the derived group or the commutator subgroup of G. Note that one must consider the subgroup generated by the set of commutators because in general the set of commutators is not closed under the group operation. Commutators are used to define nilpotent and solvable groups.

The above definition of the commutator is used by some group theorists, as well as throughout this article. However, many other group theorists define the commutator as

Identities (group theory)

Commutator identities are an important tool in group theory. The expression a^x denotes the conjugate of a by x, defined as x⁻¹a x.

Plugboard

A plugboard, or control panel (the term used depended on the application area), is an array of jacks, or sockets (often called hubs), into which patch cords can be inserted to complete an electrical circuit. Control panels were used to direct the operation of some unit record equipment. Plugboards were used on some cipher machines, and some early computers.

Unit record equipment

The earliest machines were hard wired for specific applications. Control panels were introduced in 1906 for the Hollerith Type 1 Tabulator (photo of Type 3 with built-in control panel here). Removable control panels were introduced with the Hollerith (IBM) type 3-S tabulator in the 1920s. Applications then could be wired on separate control panels, and inserted into tabulators as needed. Removable control panels came to be used in all unit record machines where the machines use for different applications required rewiring.

IBM removable control panels ranged in size from 6 1/4" by 10 3/4" (for machines such as the IBM 077, IBM 550, IBM 514) to roughly one to two feet (300 to 600 mm) on a side and had a rectangular array of hubs.Plugs at each end of a single-conductor patch cord were inserted into hubs, making a connection between two contacts on the machine when the control panel was placed in the machine, thereby connecting an emitting hub to an accepting or entry hub. For example, in a card duplicator application a card column reading (emitting) hub might be connected to a punch magnet entry hub. It was a relatively simple matter to copy some fields, perhaps to different columns, and ignore other columns by suitable wiring. Tabulator control panels could require dozens of patch cords for some applications.

Commutator (electric)

A commutator is a moving part of a rotary electrical switch in certain types of electric motors and electrical generators that periodically reverses the current direction between the rotor and the external circuit. It consists of a cylinder composed of multiple metal contact segments on the rotating armature of the machine. The commutator is one component of a motor; there are also two or more stationary electrical contacts called "brushes" made of a soft conductor like carbon that press against the commutator, making sliding contact with successive segments of the commutator as it rotates. The windings (coils of wire) on the armature are connected to the commutator segments.

Commutators are used in direct current (DC) machines: dynamos (DC generators) and many DC motors as well as universal motors. In a motor the commutator applies electric current to the windings. By reversing the current direction in the rotating windings each half turn, a steady rotating force (torque) is produced. In a generator the commutator picks off the current generated in the windings, reversing the direction of the current with each half turn, serving as a mechanical rectifier to convert the alternating current from the windings to unidirectional direct current in the external load circuit. The first direct current commutator-type machine, the dynamo, was built by Hippolyte Pixii in 1832, based on a suggestion by André-Marie Ampère.