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The subgroup of order n / d is a subgroup of the subgroup of order n / e if and only if e is a divisor of d. The lattice of subgroups of the infinite cyclic group can be described in the same way, as the dual of the divisibility lattice of all positive integers. If the infinite cyclic group is represented as the additive group on the integers ...
A cyclic group is a group which is equal to one of its cyclic subgroups: G = g for some element g, called a generator of G. For a finite cyclic group G of order n we have G = {e, g, g2, ... , gn−1}, where e is the identity element and gi = gj whenever i ≡ j (mod n); in particular gn = g0 = e, and g−1 = gn−1.
In mathematics, specifically group theory, the index of a subgroup H in a group G is the number of left cosets of H in G, or equivalently, the number of right cosets of H in G. The index is denoted or or . Because G is the disjoint union of the left cosets and because each left coset has the same size as H, the index is related to the orders of ...
The symmetric group on a finite set is the group whose elements are all bijective functions from to and whose group operation is that of function composition. [1] For finite sets, "permutations" and "bijective functions" refer to the same operation, namely rearrangement. The symmetric group of degree is the symmetric group on the set .
Subgroup. In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.
In group theory, a subfield of abstract algebra, a cycle graph of a group is an undirected graph that illustrates the various cycles of that group, given a set of generators for the group. Cycle graphs are particularly useful in visualizing the structure of small finite groups. A cycle is the set of powers of a given group element a, where an ...
Lattice of subgroups. In mathematics, the lattice of subgroups of a group is the lattice whose elements are the subgroups of , with the partial ordering being set inclusion. In this lattice, the join of two subgroups is the subgroup generated by their union, and the meet of two subgroups is their intersection.
The Schur multiplier of a finite group G is a finite abelian group whose exponent divides the order of G. If a Sylow p -subgroup of G is cyclic for some p, then the order of is not divisible by p. In particular, if all Sylow p -subgroups of G are cyclic, then is trivial. For instance, the Schur multiplier of the nonabelian group of order 6 is ...