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related to: another word for group of three objects found in the following series
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In mathematics, specifically group theory, a subgroup series of a group is a chain of subgroups: = = where is the trivial subgroup.Subgroup series can simplify the study of a group to the study of simpler subgroups and their relations, and several subgroup series can be invariantly defined and are important invariants of groups.
A starting point for the Fischer groups is the unitary group PSU 6 (2), which could be thought of as a group Fi 21 in the series of Fischer groups, of order 9,196,830,720 = 2 15 ⋅3 6 ⋅5⋅7⋅11. Actually it is the double cover 2.PSU 6 (2) that becomes a subgroup of the new group.
For example, the symmetric group on 3 letters is the group of all possible reorderings of the objects. The three letters ABC can be reordered into ABC, ACB, BAC, BCA, CAB, CBA, forming in total 6 (factorial of 3) elements. The group operation is composition of these reorderings, and the identity element is the reordering operation that leaves ...
The group consists of the finite strings (words) that can be composed by elements from A, together with other elements that are necessary to form a group. Multiplication of strings is defined by concatenation, for instance (abb) • (bca) = abbbca. Every group (G, •) is basically a factor group of a free group generated by G.
An algebraic group is a group object in the category of algebraic varieties. In modern algebraic geometry, one considers the more general group schemes, group objects in the category of schemes. A localic group is a group object in the category of locales. The group objects in the category of groups (or monoids) are the abelian groups.
There is a common denominator: a group of data objects. However, different researchers employ different cluster models, and for each of these cluster models again different algorithms can be given. The notion of a cluster, as found by different algorithms, varies significantly in its properties.
The Three Hunters, Aragorn, Legolas, and Gimli, are central figures who proved that point when a human, elf prince, and dwarf warrior (respectively) found strength in their diverse backgrounds.
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object.