Symmetric group

In abstract algebra, the symmetric group S_n on a finite set of n symbols is the group whose elements are all the permutation operations that can be performed on n distinct symbols, and whose group operation is the composition of such permutation operations, which are defined as bijective functions from the set of symbols to itself. Since there are n! (n factorial) possible permutation operations that can be performed on a tuple composed of n symbols, it follows that the order (the number of elements) of the symmetric group S_n is n!.

Although symmetric groups can be defined on infinite sets as well, this article discusses only the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set.

The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the representation theory of Lie groups, and combinatorics. Cayley's theorem states that every group G is isomorphic to a subgroup of the symmetric group on G.

Group

Group may refer to:

Groups of people

In science and technology

In mathematics

In chemistry

In computing and the internet

Other uses in science and technology

Other uses

Shot grouping

In shooting, a shot grouping, or simply grouping, is the placement of multiple shots on a target, the shots taken in one shooting session. The closeness of the grouping, the nearness of all the shots to each other, is a measure of the accuracy of a weapon, and a measure of the shooter's consistency and skill.

Uses of the term

For firearms that shoot one round at a time, a shot grouping can be used to measure the accuracy of the system comprised out of weapon as well as the precision and uniformity of the ammunition by fixing the weapon into position on a test mount, and aiming it at a target. Multiple shots using rounds from same type and batch are fired to observe how the weapon groups the shots. If a person holds the weapon and shoots it, the grouping measures the combination of the person's skill and the weapon's accuracy.

In shotgun shooting, the grouping is also called the pattern. The pattern is the spread of shot from a single shotgun shell, measured as the smallest circle containing all the shots on the target. The barrel of a shotgun is designed to deliver a wide or narrow grouping, depending on the expected use. Shooting at close range indicates a cylinder bore barrel to deliver a wide grouping, while for hunting at longer distances such as 50 yards or meters, a choke is recommended for a tighter grouping.

Stratigraphic unit

A stratigraphic unit is a volume of rock of identifiable origin and relative age range that is defined by the distinctive and dominant, easily mapped and recognizable petrographic, lithologic or paleontologic features (facies) that characterize it.

Units must be mappable and distinct from one another, but the contact need not be particularly distinct. For instance, a unit may be defined by terms such as "when the sandstone component exceeds 75%".

Lithostratigraphic units

Sequences of sedimentary and volcanic rocks are subdivided on the basis of their lithology. Going from smaller to larger in scale, the main units recognised are Bed, Member, Formation, Group and Supergroup.

Bed

A bed is a lithologically distinct layer within a member or formation and is the smallest recognisable stratigraphic unit. These are not normally named, but may be in the case of a marker horizon.

Member

A member is a named lithologically distinct part of a formation. Not all formations are subdivided in this way and even where they are recognized, they may only form part of the formation.