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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 .
In mathematics, especially in the area of algebra known as group theory, the term Z-group refers to a number of distinct types of groups: in the study of finite groups, a Z-group is a finite group whose Sylow subgroups are all cyclic. in the study of infinite groups, a Z-group is a group which possesses a very general form of central series.
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 exponent p for p > 2. Order p 4: The classification is complicated, and gets much harder as the exponent of p increases.
A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, H ≠ G). This is often represented notationally by H < G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (that is, H ≠ {e} ). [2] [3] If H is a subgroup of G, then G is sometimes called an overgroup of H.
Hence, the fundamental group of the Cayley graph Γ(G) is isomorphic to the kernel of φ, the normal subgroup of relations among the generators of G. The extreme case is when G = {e}, the trivial group, considered with as many generators as F, all of them trivial; the Cayley graph Γ(G) is a bouquet of circles, and its fundamental group is F ...
The principal congruence subgroup of level 2, Γ(2), is also called the modular group Λ. Since PSL(2, Z/2Z) is isomorphic to S 3, Λ is a subgroup of index 6. The group Λ consists of all modular transformations for which a and d are odd and b and c are even.
If we then let N be the subgroup of F generated by all conjugates x −1 Rx of R, then it follows by definition that every element of N is a finite product x 1 −1 r 1 x 1... x m −1 r m x m of members of such conjugates. It follows that each element of N, when considered as a product in D 8, will also evaluate to 1; and thus that N is a ...
It is possible to define a graph with vertices and edges the disjoint union of all coset spaces Γ/G x and Γ/G y respectively. This graph is a tree, called the universal covering tree, on which Γ acts. It admits the graph Y as fundamental domain. The graph of groups given by the stabilizer subgroups on the fundamental domain corresponds to ...