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  2. Lattice of subgroups - Wikipedia

    en.wikipedia.org/wiki/Lattice_of_subgroups

    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.

  3. Lattice (discrete subgroup) - Wikipedia

    en.wikipedia.org/wiki/Lattice_(discrete_subgroup)

    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).

  4. Lattice (group) - Wikipedia

    en.wikipedia.org/wiki/Lattice_(group)

    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}} .

  5. Subgroup - Wikipedia

    en.wikipedia.org/wiki/Subgroup

    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.

  6. List of finite simple groups - Wikipedia

    en.wikipedia.org/wiki/List_of_finite_simple_groups

    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.

  7. Subgroups of cyclic groups - Wikipedia

    en.wikipedia.org/wiki/Subgroups_of_cyclic_groups

    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 ...

  8. Parabolic subgroup of a reflection group - Wikipedia

    en.wikipedia.org/wiki/Parabolic_subgroup_of_a...

    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.

  9. Normal subgroup - Wikipedia

    en.wikipedia.org/wiki/Normal_subgroup

    A concrete example of a normal subgroup is the subgroup = {(), () ... , in this lattice is their intersection and the join is their product. The lattice ...