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For example, the collective noun "group" can be applied to people ("a group of people"), or dogs ("a group of dogs"), or objects ("a group of stones"). Some collective nouns are specific to one kind of thing, especially terms of venery, which identify groups of specific animals. For example, "pride" as a term of venery always refers to lions ...
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 ...
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.
When comparing the symmetry type of two objects, the origin is chosen for each separately, i.e., they need not have the same center. Moreover, two objects are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of O(3) (two subgroups H 1, H 2 of a group G are conjugate, if there exists g ∈ G such that H 1 = g −1 H 2 g).
The group had more than three members at various points, but these three were the last ones standing. ... Patterns can be the shapes of animals, objects, plants, or even people. Three is the ...
The zero objects in Grp are the trivial groups (consisting of just an identity element). Every morphism f : G → H in Grp has a category-theoretic kernel (given by the ordinary kernel of algebra ker f = { x in G | f ( x ) = e }), and also a category-theoretic cokernel (given by the factor group of H by the normal closure of f ( G ) in H ).
Think of the group of three friendship like a plant: It must be well-watered. And that means recognizing there are four relationships at play: A/B, A/C, B/C and the collective A/B/C.
The word syzygy is often used to describe interesting configurations of astronomical objects in general. For example, one such case occurred on March 21, 1894, around 23:00 GMT , when Mercury transited the Sun as would have been seen from Venus, and Mercury and Venus both simultaneously transited the Sun as seen from Saturn .