Search results
Results from the WOW.Com Content Network
The lattice formed by these ten subgroups is shown in the illustration. This example also shows that the lattice of all subgroups of a group is not a modular lattice in general. Indeed, this particular lattice contains the forbidden "pentagon" N 5 as a sublattice.
Let be a locally compact group and a discrete subgroup (this means that there exists a neighbourhood of the identity element of such that = {}).Then is called a lattice in if in addition there exists a Borel measure on the quotient space / which is finite (i.e. (/) < +) and -invariant (meaning that for any and any open subset / the equality () = is satisfied).
A simple example of a lattice in is the subgroup . More complicated examples include the E8 lattice , which is a lattice in R 8 {\displaystyle \mathbb {R} ^{8}} , and the Leech lattice in R 24 {\displaystyle \mathbb {R} ^{24}} .
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.
The perfect double cover Co 0 of Co 1 is the automorphism group of the Leech lattice, and is sometimes denoted by ·0. Subgroup of Co 0; fixes a norm 4 vector in the Leech lattice. Subgroup of Co 0; fixes a norm 6 vector in the Leech lattice. It has a doubly transitive permutation representation on 276 points.
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, then the subgroup generated by d is a subgroup of the subgroup generated by e if and only if e is a divisor ...
The lattice of parabolic subgroups of the dihedral group D 2×4, represented as a real reflection group, consists of the trivial subgroup, the four two-element subgroups generated by a single reflection, and the entire group. Ordered by inclusion, they give the same lattice as the lattice of fixed spaces ordered by reverse-inclusion.
A concrete example of a normal subgroup is the subgroup = {(), () ... , in this lattice is their intersection and the join is their product. The lattice ...