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The symmetry group of a cube is the internal direct product of the subgroup of rotations and the two-element group {−I, I}, where I is the identity element and −I is the point reflection through the center of the cube. A similar fact holds true for the symmetry group of an icosahedron. Let n be odd, and let D 4n be the dihedral group of ...
A presentation of a group determines a geometry, in the sense of geometric group theory: one has the Cayley graph, which has a metric, called the word metric. These are also two resulting orders, the weak order and the Bruhat order, and corresponding Hasse diagrams. An important example is in the Coxeter groups.
Order p: The only group is cyclic. Order p 2: There are just two groups, both abelian. Order p 3: There are three abelian groups, and two non-abelian groups. One of the non-abelian groups is the semidirect product of a normal cyclic subgroup of order p 2 by a cyclic group of order p. The other is the quaternion group for p = 2 and a group of ...
There are only two cosets: the set of even integers and the set of odd integers, and therefore the quotient group / is the cyclic group with two elements. This quotient group is isomorphic with the set {,} with addition modulo 2; informally, it is sometimes said that / equals the set {,} with addition modulo 2.
The number of possible person-to-person links (L) increases rapidly as the size of the group (N) increases (L = (N² - N) /2). In a four-member group there are six possible pairings; add a fifth member for each of the four to relate to and you have ten pairs. The number of possible two-person links in a group of twelve is 66.
The 5th roots of unity in the complex plane form a group under multiplication. Each non-identity element generates the group. In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses.
Create distribution lists to save time when you send emails to a group of contacts from the contacts you already have in your AOL Contacts, set up a contact list with a group of people you often send emails. For example, you email the same content to 3 friends every week. Instead, create a contact list called "Friends".
A consequence of the theorem is that the order of any element a of a finite group (i.e. the smallest positive integer number k with a k = e, where e is the identity element of the group) divides the order of that group, since the order of a is equal to the order of the cyclic subgroup generated by a. If the group has n elements, it follows